Analysis of approximation by linear operators on variable L ρ p (middot) spaces and applications inlearning theory

Li Bing-Zheng Zhou Ding-Xuan

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Research ArticleAnalysis of Approximation by Linear Operators on Variable 119871119901(sdot)

120588

Spaces and Applications in Learning Theory

Bing-Zheng Li1 and Ding-Xuan Zhou2

1 Department of Mathematics Zhejiang University Hangzhou 310027 China2Department of Mathematics City University of Hong Kong Kowloon Hong Kong

Correspondence should be addressed to Ding-Xuan Zhou mazhoucityueduhk

Received 7 May 2014 Accepted 30 June 2014 Published 16 July 2014

Academic Editor Uno Hamarik

Copyright copy 2014 B-Z Li and D-X Zhou This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper is concerned with approximation on variable 119871119901(sdot)

120588spaces associated with a general exponent function 119901 and a general

bounded Borel measure 120588 on an open subset Ω of R119889 We mainly consider approximation by Bernstein type linear operatorsUnder an assumption of log-Holder continuity of the exponent function 119901 we verify a conjecture raised previously about theuniform boundedness of Bernstein-Durrmeyer and Bernstein-Kantorovich operators on the 119871

119901(sdot)

120588space Quantitative estimates

for the approximation are provided for high orders of approximation by linear combinations of such positive linear operatorsMotivating connections to classification and quantile regression problems in learning theory are also described

1 Introduction

Approximation by Bernstein type positive linear operatorshas a long history and is an important topic in approximationtheory It started with Bernstein operators [1] for provingthe Weierstrass theorem about the denseness of the set ofpolynomials in the space 119862[0 1] of continuous functions onthe interval [0 1] These classical operators are defined as119861119899(119891 119909) = sum

119899

119896=0119891(119896119899)119901119899119896(119909) for 119909 isin [0 1] and 119891 isin 119862[0 1]

with the Bernstein basis given by 119901119899119896(119909) = (119899119896 ) 119909

119896(1 minus 119909)

119899minus119896The Bernstein operators have been extended in various formsfor the purpose of approximating discontinuous functions byreplacing the point evaluation functionals by some integralsThe classical examples for approximation in 119871

119901[0 1] (with

1 le 119901 lt infin) the Banach space of all integrable functions

119891 on [0 1] with the norm 119891119871119901

= (int1

0|119891(119909)|

119901119889119909)

1119901

areBernstein-Kantorovich operators [2]

119870119899 (119891 119909)

=

119899

sum

119896=0

(119899 + 1) int

(119896+1)(119899+1)

119896(119899+1)

119891 (119905) 119889119905119901119899119896 (119909) 119909 isin [0 1](1)

and Bernstein-Durrmeyer operators [3]

119863119899 (119891 119909) =

119899

sum

119896=0

(119899 + 1) int

1

0

119901119899119896 (119905) 119891 (119905) 119889119905119901119899119896 (119909)

119909 isin [0 1]

(2)

Quantitative estimates for approximation by Bernstein typepositive linear operators in 119862[0 1] or 119871

119901[0 1] have been

presented in a large literature (eg [4 5]) See the book [6]and references therein for details and extensions to infiniteintervals and linear combinations of positive operators forachieving high orders of approximation

In this paper we provide a general framework for approx-imation by linear operators on variable 119871

119901(sdot)

120588(Ω) spaces on an

open subset Ω of R119889 Here 119901 Ω rarr [1 infin) is a measurablefunction called the exponent function and 120588 is a positivebounded Borel measure on Ω The variable space 119871

119901(sdot)

120588(Ω) is

a generalization of the weighted 119871119901 spaces with a constant

exponent 119901 isin [1 infin] It consists of all the measurable

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 454375 10 pageshttpdxdoiorg1011552014454375

2 Abstract and Applied Analysis

functions119891 onΩ such thatintΩ

(|119891(119909)|120582)119901(119909)

119889120588 lt infin for some120582 gt 0 The norm is defined by

10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

= inf 120582 gt 0 intΩ

(

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 le 1 (3)

The space 119871119901(sdot)

120588(Ω) is a Banach space [7] The idea of

variable 119871119901(sdot) spaces was introduced by Orlicz [8] Motivated

by connections to variational integrals with nonstandardgrowth related to modeling of electrorheological fluids [9]these function spaces have been developed in analysis andresearch topics include boundedness of maximal operatorscontinuity of translates and denseness of smooth functionsWe will not go into details which can be found in [710] and references therein Instead we only mention thefollowing core condition on the log-Holder continuity of theexponent function which leads to the boundedness of Hardy-Littlewood maximal operators and the rich theory of thevariable 119871

119901(sdot)

120588(Ω) spaces

Definition 1 We say that the exponent function 119901 Ω rarr

[1 infin) is log-Holder continuous if there exist positive con-stants 119860119901 gt 0 such that

1003816100381610038161003816119901 (119909) minus 119901 (119910)1003816100381610038161003816 le

119860119901

minus log 1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119909 119910 isin Ω1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 lt1

2

(4)

We say that 119901 is log-Holder continuous at infinity (when Ω isunbounded) if there holds

1003816100381610038161003816119901 (119909) minus 119901 (119910)1003816100381610038161003816 le

119860119901

log (119890 + |119909|) 119909 119910 isin Ω

10038161003816100381610038161199101003816100381610038161003816 ge |119909| (5)

Denote

119901minus = inf119909isinΩ

119901 (119909) 119901+ = sup119909isinΩ

119901 (119909) (6)

The issue of approximation by Bernstein type positivelinear operators on variable 119871

119901(sdot)

120588(Ω) spaces was raised by the

second author in [11] It turned out that the variety of theexponent function 119901 creates technical difficulty in the studyof approximation In particular the uniform boundednessof the Bernstein-Kantorovich operators (1) and Bernstein-Durrmeyer operators (2) is already a difficult problem Thekey analysis in [11] is to show that the Bernstein-Kantorovichoperators and Bernstein-Durrmeyer operators are uniformlybounded when the exponent function 119901 is Lipschitz 120572 forsome 120572 isin (0 1] It was conjectured there that the uniformboundedness still holds when 119901 is log-Holder continuousThe firstmain result of this paper is to confirm this conjectureinTheorem 6 below

Our second main result is to abandon the positivity andpresent quantitative estimates for the high order approxi-mation by linear operators including linear combinations ofBernstein type positive linear operators extending the resultsin [11] for the first order approximation by positive operators

2 Motivations from Learning Theory

Our main motivation for considering the approximation offunctions by linear operators on variable 119871

119901(sdot)

120588(Ω) spaces is

from learning theory Besides the example of extending theBernstein-Durrmeyer operators (2) to those associated witha general probability measure 120588 on Ω in [12 13] for themultivariate case we mention two learning theory settingshere Since error analysis for concrete learning algorithmsin terms of the introduced noise conditions involves sampleerror estimates which are out of the scope of this paper weleave the detailed error bounds to our further study

21 Noise Conditions for Classification and ApproximationThe first learning theory setting related to approximationon variable 119871

119901(sdot)

120588(Ω) spaces is noise conditions for binary

classification Here Ω is an input space consisting of possibleevents while the output space is denoted as 119884 = 1 minus1A Borel probability measure 119875 on the product space Ω times 119884

can be decomposed into its marginal distribution 119875Ω on Ω

and conditional distributions 119875(sdot | 119909) for 119909 isin Ω A binaryclassifier 119891 Ω rarr 119884 makes predictions 119891(119909) isin 119884 for futureevents 119909 isin Ω The best classifier 119891119888 called Bayes rule is givenby 119891119888(119909) = 1 if 119875(1 | 119909) gt 12 and minus1 otherwise Theprobability measure 119875 fits the binary classification problemwell if the conditional probabilities 119875(1 | 119909) and 119875(minus1 | 119909)

are well separated from the boundary 12 for most events119909 Their separations are equivalent to the separation of thevalue 119891119875(119909) of the regression function 119891119875(119909) = int

119884119910119889119875(119910 |

119909) = 119875(1 | 119909) minus 119875(minus1 | 119909) from 0 and can be measured invarious quantitative ways The Tsybakov noise condition [14]with noise exponent 119902 isin (0 infin] asserts that for some constant119888119902 gt 0 there holds

119875Ω (119909 isin Ω 0 lt1003816100381610038161003816119891119875 (119909)

1003816100381610038161003816 le 119888119902119905) le 119905119902 forall119905 gt 0 (7)

When 119902 = infin Tsybakov noise condition (7) means |119891119875(119909)| ge

119888119902 almost surely and 119891119875(119909) is well separated from 0 Thecase 119902 lt infin means the measure of the set of events 119909

with 119891119875(119909) not well separated from 0 decays polynomiallyfast as the threshold 119888119902119905 tends to 0 More details about theTsybakov noise condition the so-called Tsybakov functionand its applications to the study of classification problems canalso be found in [15] Here we introduce a noise conditionby allowing some noise situations measured by an exponentfunction 119901

Example 2 We say that the probability measure 119875 satisfiesthe noise condition associated with an exponent function119901 Ω rarr [0 infin) if for some 120582 gt 0 there holds

intΩ

(

1003816100381610038161003816119891119875 (119909)1003816100381610038161003816

120582)

119901(119909)

119889119875Ω lt infin (8)

Remark 3 The above condition can be applied to the regres-sion setting for dealingwith unbounded regression functionsWhen 119901 takes values on [1 infin) the above condition isequivalent to the requirement 119891119875 isin 119871

119901(sdot)

120588(Ω) with 120588 = 119875Ω

Abstract and Applied Analysis 3

When 119889 = 1 and Ω = R we apply the classical identityintR

119892(119909)119889119875Ω = intinfin

0119875Ω(119909 isin Ω 119892(119909) ge 119905)119889119905 to the

nonnegative function 119892(119909) = (|119891119875(119909)|120582)119901(119909) and find that the

above condition is equivalent to

int

infin

0

119875Ω (119909 isin Ω 1003816100381610038161003816119891119875 (119909)

1003816100381610038161003816 ge 1205821199051119901(119909)

) 119889119905 lt infin (9)

This illustrates some similarity between the noise condition(8) and Tsybakov noise condition (7)

The following is an example to show some differences

Example 4 Let 119889 isin N and Ω = 119909 isin R119889 |119909| le 12 If 119875Ω is

the normalized Lebesgue measure on Ω and 119891119875(119909) = 119890minus1|119909|

then the measure 119875 satisfies the noise condition associatedwith the exponent function119901(119909) = |119909| but does not satisfy theTsybakov noise condition (7) with any 119902 isin (0 infin] In fact wehave int

Ω|119891119875(119909)|

119901(119909)119889119875Ω = 1119890 lt infin while for any 119902 isin (0 infin)

and 119888119902 gt 0 we have 119875Ω(119909 isin Ω 0 lt |119891119875(119909)| le 119888119902119905) =

(2 minus log(119888119902119905))119889

gt 119905119902 for 119905 isin (0min119905

lowast 1119888119902119890

2) with 119905

lowast beingthe positive solution to the equation 119905

119902119889 log 1119888119902119905 = 2

22 Noise Conditions for Quantile Regression and Approxima-tion The second learning theory setting related to approx-imation on variable 119871

119901(sdot)

120588(Ω) spaces is noise conditions for

quantile regression Here the output space is 119884 = R Similarto the least squares regression [16] for learning means ofconditional distributions 119875(sdot | 119909) but providing richerinformation [17] about response variables such as stretchingor compressing tails the learning problem for quantile regres-sion aims at estimating quantiles of conditional distributionsWith a quantile parameter 0 lt 120591 lt 1 the value of a quantileregression function119891119875120591 at119909 isin Ω is defined by its value119891119875120591(119909)

as a 120591-quantile of 119875(sdot | 119909) that is a value 119905lowast

isin 119884 satisfying

119875 (119910 isin 119884 119910 le 119905lowast | 119909) ge 120591

119875 (119910 isin 119884 119910 ge 119905lowast | 119909) ge 1 minus 120591

(10)

Quantile regression has been studied by kernel-based regu-larization schemes in a learning theory literature (eg [1819]) For optimal error analysis of these learning algorithmsasymptotic behaviors of the conditional distributions near the120591-quantiles are needed In particular one is interested in howslow the following function decays as 119905 decreases

119865119875120591 (119909) = min 119875 (119910 isin 119884 119905lowast

minus 119905 lt 119910 lt 119905lowast | 119909)

119875 (119910 isin 119884 119905lowast

+ 119905 gt 119910 gt 119905lowast | 119909)

(11)

A noise condition was introduced in [18] by requiring lowerbounds 119865119875120591(119909) ge 119887119909119905

119902minus1 for every 119905 isin [0 119886119909] and some119902 isin (1 infin) 119901 isin (0 infin) and constants 119887119909 119886119909 gt 0

satisfying (119887119909119886119902minus1

119909)minus1

isin 119871119901

119875Ω This condition was extended to

a logarithmic bound in [19] by replacing 119905119902minus1 by (log(1119905))

minus119902

and 119886119902minus1

119909by (log(1119886119909))

minus119902 Here we introduce the followingnoise condition which is more general than the one in [18] byallowing the indices 119902 119901 to depend on the events 119909 isin Ω

Example 5 We say that the probabilitymeasure119875 satisfies thequantile noise condition associated with exponent functions119902 Ω rarr (1 infin) and 119901 Ω rarr (0 infin) if for every 119909 isin 119883there exist a 120591-quantile 119905

lowastisin R and constants 119886119909 isin (0 2] 119887119909 gt

0 such that for each 119906 isin [0 119886119909]

119865119875120591 (119909) ge 119887119909119905119902(119909)minus1 (12)

and that for some 120582 gt 0 there holdsintΩ

(1120582119887119909119886119902(119909)minus1

119909)119901(119909)

119889119875Ω lt infin

While the lower bounds (12) imply polynomial decaysof the conditional distributions near the 120591-quantiles with apower index depending on the event the finiteness of theintegral is equivalent to the requirement that the function1119887119909119886

119902(119909)minus1

119909lies in the variable 119871

119901(sdot)

119875Ω(Ω) space (when 119901 takes

values in [1 infin))

3 Main Results for Approximation on 119871119901(sdot)

120588

Our first theorem is about the uniform boundedness of asequence of linear operators on the variable119871

119901(sdot)

120588spacesThese

operators take the form

119871119899 (119891 119909) = intΩ

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905) 119909 isin Ω 119891 isin 119871119901(sdot)

120588 (13)

in terms of their kernels 119870119899(119909 119905)infin

119899=1defined on Ω times Ω We

assume that the kernels satisfy the following three conditionswith some positive constants 1198620 ge 1 119887 119862119887 and 119862119903

(depending on 119903 isin N)

sup119905isinΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119909) le 1198620

sup119909isinΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) le 1198620

(14)

sup119909119905isinΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 le 119862119887119899

119887 forall119899 isin N (15)

intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909) le 119862119903119899

minus119903

forall119899 isin N 119903 isin N

(16)

Then the uniform boundedness follows which will beproved in Section 5

Theorem 6 Let Ω sube R119889 be an open set and an exponentfunction 119901 Ω rarr (1 infin) satisfy 1 lt 119901minus lt 119901+ lt infin and thelog-Holder continuity condition (4) If the kernels 119870119899(119909 119905)

infin

119899=1

satisfy conditions (14) (15) and (16) then the operators 119871119899infin

119899=1

on 119871119901(sdot)

120588defined by (13) are uniformly bounded as

1003817100381710038171003817119871119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (17)

by a positive constant 119872119901119887 (depending on 119901 and the constantsin (14) (15) and (16) given explicitly in the proof)

4 Abstract and Applied Analysis

Our second theorem gives orders of approximation whenthe approximated function has some smoothness stated interms of a K-functional Define a Holder space with index119903 isin N on Ω by

119882119903infin

119901= 119892 isin 119871

119901(sdot)

1205881003817100381710038171003817119892

1003817100381710038171003817119901119903infin lt infin (18)

where 119892119901119903infin

is the norm given by 119892119901119903infin

= 119892119871119901(sdot)

120588

+

sum|120572|1le119903119863

120572119892

infinwith 119863

120572119892 = 120597

|120572|112059711990912057211

sdot sdot sdot 120597119909120572119889

119889and |120572|1 =

sum119889

119895=1120572119895 for 120572 = (1205721 120572119889) isin Z119889

+ The K-functional

K119903(119891 119905)119901(sdot) is defined by

K119903(119891 119905)119901(sdot)

= inf119892isin119882119903infin119901

1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588+ 119905

10038171003817100381710038171198921003817100381710038171003817119901119903infin 119905 gt 0

(19)

Denote 119862infin

0(Ω) as the space of all compactly supported

119862infin functions on Ω From [7] we know that when 119901

+lt

infin 119862infin

0(Ω) is dense in 119871

119901(sdot)

120588(Ω) Hence for any 119891 isin 119871

119901(sdot)

120588(Ω)

there holdsK119903(119891 119905)119901(sdot) rarr 0 as 119905 rarr 0The following theorem to be proved in Section 5 and

extending the results for 119903 = 1 in [11] gives orders ofapproximation by linear operators on 119871

119901(sdot)

120588(Ω) when the K-

functional has explicit decay rates

Theorem 7 Under the assumption of Theorem 6 if Ω isconvex 119903 isin N and the kernels satisfy 119903119901minus ge 2 and

intΩ

119870119899 (119909 119905) (119905 minus 119909)120572119889120588 (119905) = 1205751205720 forall120572 isin Z

119889119908119894119905ℎ |120572|1 lt 119903

(20)

for almost every 119909 isin Ω then there holds for any 119891 isin 119871119901(sdot)

120588

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) (21)

where 119903minus is the integer part of 119903119901minus2 and the constant 119860119901119887119889 isindependent of 119891 isin 119871

119901(sdot)

120588(given explicitly in the proof)

The vanishing moment assumption (20) corresponds toStrang-Fix type conditions in the literature of shift-invariantspaces for example [20 21] It has appeared in the literatureof Bernstein type operators when linear combinations areconsidered as described by (34) in the next section

4 Approximation by Bernstein Type Operators

In this section we apply our main results to Bernstein typepositive linear operators and give high orders of approxima-tion by linear combinations of these operators on variable119871119901(sdot)

120588(Ω) spaces We demonstrate the analysis for the general

Bernstein-Durrmeyer operators in detail and describe brieflyresults for the general Bernstein-Kantorovich operators as anexample of other families of operators

The Bernstein-Durrmeyer operators on an open simplex

Ω = 119878 = 119909 isin R119889

+ |119909|1 lt 1 (22)

associated with a general positive Borel measure 120588 on 119878 aredefined as

119863119899 (119891 119909) = sum

|120572|1le119899

int119878

119891 (119905) 119901119899120572 (119905) 119889120588 (119905)

int119878

119901119899120572 (119905) 119889120588 (119905)119901119899120572 (119909)

119891 isin 1198711

120588 119909 isin 119878

(23)

where for 119909 = (1199091 119909119889) isin 119878 sub R119889 120572 = (1205721 120572119889) isin Z119889

+

we denote

119901119899120572 (119909) =119899

Π119889119895=1

120572119895 (119899 minus |120572|1)

119889

prod

119895=1

119909120572119895

119895(1 minus |119909|1)

119899minus|120572|1 (24)

The classical Bernstein-Durrmeyer operators (2) on Ω =

(0 1) (119889 = 1) with 119889120588(119909) = 119889119909 have been well studied (eg[22]) and extended to a multivariate form with respect toJacobi weights 119889120588(119909) = Π

119889

119895=1119909119902119895

119895119889119909 (eg [23]) Bernstein-

Durrmeyer operators on Ω = (0 1) with respect to anarbitrary Borel probability measure were introduced in [12]and applied to error analysis of learning algorithms forsupport vectormachine classificationsThemultidimensionalversion of such linear operators (23)was introduced in [13] In[24] the first author showed for a constant exponent function119901(119909) equiv 119901 isin [1 infin) that lim119899rarrinfin119891 minus 119863119899(119891)

119871119901

120588= 0 for any

119891 isin 119871119901

120588 The case 119901 equiv infin was studied in [25 26] Here we

consider the case with a general exponent function satisfying1 lt 119901minus lt 119901+ lt infin and the log-Holder continuity condition(4)

By applyingTheorem 6we can prove the uniformbound-edness of the Bernstein-Durrmeyer operators (23)

Proposition 8 LetΩ = 119878 120588 be a Borel probability measure on119878 and an exponent function 119901 119878 rarr (1 infin) satisfy 1 lt 119901minus lt

119901+ lt infin and the log-Holder continuity condition (4) If thereexist positive constants 119887 and 119862

lowast

119887such that int

119878119901119899120572(119905)119889120588(119905) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Durrmeyer operators defined on 119871119901(sdot)

120588(Ω) by (23) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119863119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (25)

Proof Define a sequence of kernels Ψ119899 on 119878 times 119878 by

Ψ119899 (119909 119905) = sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905) (26)

Then the Bernstein-Durrmeyer operators (23) can be writtenas

119863119899 (119891 119909) = int119878

Ψ119899 (119909 119905) 119891 (119905) 119889120588 (119905) (27)

So we only need to check the three conditions (14) (15) and(16) of Theorem 6

Abstract and Applied Analysis 5

Since Ψ119899(119909 119905) ge 0 we know that

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 119889120588 (119909) = int

119878

Ψ119899 (119909 119905) 119889120588 (119909)

= sum

|120572|1le119899

119901119899120572 (119905) int119878

119901119899120572 (119909) 119889120588 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

= sum

|120572|1le119899

119901119899120572 (119905) equiv 1

(28)

The same is true for int119878

|Ψ119899(119909 119905)|119889120588(119905) equiv 1 So we know thatcondition (14) holds with the constant 1198620 = 1

Applying the lower bound int119878

119901119899120572(119905)119889120588(119905) ge 119862lowast

119887119899minus119887 and

the inequality 119901119899120572(119905) le 1 we see that for any 119899 isin N and119909 119905 isin 119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 le sum

|120572|1le119899

119901119899120572 (119909)

119862lowast

119887119899minus119887

=1

119862lowast

119887

119899119887 (29)

Hence condition (15) holds with 119862119887 = 1119862lowast

119887

As for the last condition we separate 119905 minus 119909 into 119905 minus (120572119899) +

(120572119899) minus 119909 and find that for any 119903 isin N and 119899 isin N

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816|119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909)

le 22119903

int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

+ int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

= 22119903

int119878

sum

|120572|1le119899

119901119899120572 (119905)1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905)

+ int119878

sum

|120572|1le119899

119901119899120572 (119909)1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119909)

= 22119903+1

int119878

119861119899 ((sdot minus 119909)2119903

119909) 119889120588 (119909)

(30)

where 119861119899 is the multidimensional Bernstein operators on theclosure 119878 of 119878 defined by

119861119899 (119891 119909) = sum

|120572|1le119899

119891 (120572

119899) 119901119899120572 (119909) 119909 isin 119878 119891 isin 119862 (119878) (31)

It is well known [6] for the multidimensional Bernsteinoperators that there exists a constant 119862119889119903 depending only on119889 and 119903 such that

119861119899 ((sdot minus 119909)2119903

119909) le 119862119889119903119899minus119903

forall119909 isin 119878 (32)

It follows that

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909) le 2

2119903+1119862119889119903119899

minus119903 (33)

and condition (16) holds true with 119862119903 = 22119903+1

119862119889119903With all the three conditions verified the desired uniform

bound (25) for the Bernstein-Durrmeyer operators followsfromTheorem 6 This proves the proposition

The Bernstein-Durrmeyer operators (23) are positivewhich prevent from achieving high order approximation dueto a saturation phenomenon Linear combinations of suchoperators can be used to get high orders of approximationThe idea and literature review of this method can be found in[6] while further developments will not be mentioned hereThe linear combinations are defined as

119871119899119903 (119891 119909) =

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894(119891 119909) (34)

where 119898119889119903 = (119889 + 119903 minus 1)119889(119903 minus 1) is the dimension of thespace of polynomials of degree at most 119903 minus 1 and with twopositive constants 1198611 1198612 independent of 119899 we have

119899 = 1198990 lt 1198991 lt sdot sdot sdot lt 119899119898119889119903le 1198611119899

119898119889119903

sum

119894=0

1003816100381610038161003816119862119894 (119899)1003816100381610038161003816 le 1198612

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894((sdot minus 119909)

120572 119909) = 1205751205720 forall0 le |120572|1 le 119903 minus 1

(35)

For the classical Bernstein-Durrmeyer operators with respectto the Lebesgue measure (or even the Jacobi weights) theexistence of the above linear combinations can be seenand found in the literature The existence of such linearcombinations with respect to the arbitrary measure 120588 isa nontrivial problem and deserves intensive study Thistechnical question is out of the scope of this paper and willbe discussed in our further work Here we concentrate on thevariable 119871

119901(sdot)

120588(Ω) spaces and state the following result for the

high orders of approximation under the condition (35) whichis an immediate consequence of Theorem 7

Proposition 9 Under the assumption of Proposition 8 if 2 le

119903 isin 119873 and the operators 119871119899119903119899isinN defined by (34) satisfy (35)then for any 119891 isin 119871

119901(sdot)

120588 we have

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) (36)

where 119903minus is the integer part of 119903119901minus2 and the constant 119860119901119887119889 isindependent of 119891 isin 119871

119901(sdot)

120588

Let us now briefly describe approximation results for theBernstein-Kantorovich operators on 119878 defined [27] as

119861119870119899 (119891 119909) = sum

|120572|1le119899

int119878119899120572

119891 (119905) 119889120588 (119905)

120588 (119878119899120572)119901119899120572 (119909) 119891 isin 119871

1

120588 119909 isin 119878

(37)

6 Abstract and Applied Analysis

where 119878119899120572120572 are subdomains of 119878 defined by

119878119899120572 = 119909 isin 119878 119909 isin

119889

prod

119894=1

[120572119894

119899 + 1

120572119894

119899 + 1) |119909|1 le

|120572|1 + 1

119899 + 1

|120572|1 le 119899

(38)

In the same way as for the Bernstein-Durrmeyer oper-ators we have the following results for the Bernstein-Kantorovich operators

Proposition 10 Under the assumption of Proposition 8 for 119901if there exist positive constants 119887 and 119862

lowast

119887such that 120588(119878119899120572) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Kantorovich operators defined on 119871119901(sdot)

120588(Ω) by (37) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119861119870119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (39)

If 2 le 119903 isin 119873 and with 119863119899119894replaced by 119861119870119899119894

the operators119871119899119903119899isinN defined by (34) satisfy (35) then

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) forall119891 isin 119871

119901(sdot)

120588

(40)

5 Proof of Main Results

In this section we give detailed proof of our main results Letus first proveTheorem 6

Proof of Theorem 6 Let 119891 isin 119871119901(sdot) have norm 1 which implies

intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 Choose 120574 = 119887(119901minus minus 1) gt 0For 119909 isin Ω we define two subsets Ω119899119909 and Ω

1015840

119899119909of Ω as

Ω119899119909 = 119905 isin Ω 1003816100381610038161003816119891 (119905)

1003816100381610038161003816 le 119899120574 |119905 minus 119909| le 119899

minus14

Ω1015840

119899119909= 119905 isin Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le 119899

120574 |119905 minus 119909| gt 119899

minus14

(41)

Set

1198711198991 (119891 119909) = intΩ(Ω119899119909cupΩ

1015840119899119909)

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198992 (119891 119909) = intΩ119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198993 (119891 119909) = intΩ1015840119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

(42)

Then the value 119871119899(119891 119909) can be decomposed into three partsas

119871119899 (119891 119909) = 1198711198991 (119891 119909) + 1198711198992 (119891 119909) + 1198711198993 (119891 119909) 119909 isin Ω

(43)

and we have

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 3119901+

3

sum

119895=1

intΩ

10038161003816100381610038161003816119871119899119895 (119891 119909)

10038161003816100381610038161003816

119901(119909)

119889120588 (119909)

(44)

In the followingwe estimate the three terms in (44) separately

Step 1 Estimating the First Term of (44) By the definition of119901minus we have 119901(119905) ge 119901minus gt 1 For 119905 isin Ω (Ω119899119909 cup Ω

1015840

119899119909) we have

|119891(119905)| gt 119899120574 and thereby

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le

1003816100381610038161003816119891 (119905)1003816100381610038161003816 (

1003816100381610038161003816119891 (119905)1003816100381610038161003816 119899

minus120574)119901(119905)minus1

=1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901(119905))

le1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901minus) = 119899

minus1198871003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

(45)

It follows from condition (15) that

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119899

minus119887intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905)

le 119899minus119887

119862119887119899119887

intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

(46)

So by the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119862119887 int

Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905) le 119862119887 (47)

Consequently

intΩ

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 120588 (Ω) (119862119887

119901minus+ 119862119887

119901+) (48)

Step 2 Estimating the Second Term of (44) By the conditionintΩ

|119870119899(119909 119905)|119889120588(119905) le 1198620 in (14) with 1198620 ge 1 we know by theHolder inequality

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)

119889120588 (119905)

times (intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905))

119901(119909)minus1

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

times 119889120588 (119905) 119889120588 (119909)

= 119862119901+minus1

0intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

(49)

where

119869119899 (119891 119909 119905) =1003816100381610038161003816119870119899 (119909 119905)

10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

119905 isin Ω119899119909 119909 isin Ω

(50)

For 119905 isin Ω119899119909 we have a bound |119891(119905)| le 119899120574 and the

restriction |119909 minus 119905| le 119899minus14 From the log-Holder continuity

of the exponent function 119901 there exists a constant 119860119901 onlydependent on 119901 such that

119901 (119909) minus 119901 (119905) le119860119901

minus log 119899minus14=

4119860119901

log 119899 (51)

Abstract and Applied Analysis 7

When |119891(119905)| ge 1 we find

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119909)minus119901(119905)

le (119899120574)4119860119901 log 119899

le 1198994120574119860119901 log 119899 le 119901 (52)

where the constant number 119901 defined by

119901 = sup119899isinN

1198994120574119860119901 log 119899 (53)

is finite because

log 1198994120574119860119901 log 119899 =

4120574119860119901 log 119899

log 119899997888rarr 4120574119860119901 as 119899 997888rarr infin

(54)

When |119891(119905)| lt 1 we simply use |119891(119905)|119901(119909)minus119901(119905)

le

|119891(119905)|minus119901(119905) Applying these bounds we can estimate the core

part of Step 2 as

intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

(119901 +1003816100381610038161003816119891 (119905)

1003816100381610038161003816minus119901(119905)

)

times 119889120588 (119905) 119889120588 (119909)

= 119901 intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

+ intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) 119889120588 (119909)

le 1199011198620 intΩ

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

+ 1198620120588 (Ω) le 1198620119901 + 1198620120588 (Ω)

(55)

Here we have used the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 and(14) So the second term of (44) can be estimated as

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+0

(119901 + 120588 (Ω)) (56)

Step 3 Estimating the Third Term of (44) For 119905 isin Ω1015840

119899119909 we

have |119891(119905)| le 119899120574 and |119909 minus 119905| gt 119899

minus14 yielding 11989914

|119909 minus 119905| gt 1 Itfollows that

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816 le int

Ω1015840119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119899

120574(119899

14|119909 minus 119905|)

2119903119901(119909)

119889120588 (119905)

(57)

where 119903 is the integer part of 2120574119901+ + 1 Applying the Holderinequality and (14) we see that

10038161003816100381610038161198711198993(119891 119909)1003816100381610038161003816119901(119909)

le intΩ1015840119899119909

119899120574119901(119909)+(1199032) 1003816100381610038161003816119870119899 (119909 119905)

1003816100381610038161003816 |119909 minus 119905|2119903

119889120588 (119905) 119862119901(119909)minus1

0

(58)

So by the bound 119901(119909) le 119901+ and condition (16) the third termof (44) can be bounded as

intΩ

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+minus1

0119899120574119901++(1199032)119862119903119899

minus119903

= 119862119901+minus1

0119862119903119899

120574119901+minus(1199032) le 119862119901+minus1

0119862119903

(59)

Finally we put the estimates (48) (56) and (59) into (44)to conclude

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(60)

Take

120582 = 119872119901119887 = 1 + 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(61)

we find

intΩ

(

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le (1

120582)

119901minus

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 1

(62)

This implies 119871119899(119891)119871119901(sdot)

120588

le 119872119901119887 The bound 119872119901119887 isindependent of 119891 So we have 119871119899 le 119872119901119887 The proofTheorem 6 is complete

We are now in a position to proveTheorem 7

Proof of Theorem 7 We follow the standard procedure inapproximation theory and consider the error 119871119899(119892 119909) minus 119892(119909)

for 119892 isin 119882119903infin

119901 Apply the Taylor expansion

119892 (119905) = 119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119909 119905 isin Ω

(63)

where the remainder term 119877119892119903(119909 119905) is given by

119877119892119903 (119909 119905)

= int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572(119905 minus 119909)

120572119889119906

(64)

8 Abstract and Applied Analysis

We see from the vanishing moment condition (20) that

119871119899 (119892 119909) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119889120588 (119905) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572

times (119905 minus 119909)120572119889119906 119889120588 (119905)

(65)

SinceΩ is convex 119909+119906(119905minus119909) isin Ω for any 119906 isin [0 1] 119909 119905 isin ΩSo |119863

120572119892(119909 + 119906(119905 minus 119909))| le 119892

119901119903infinand we have

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

le int

1

0

(1 minus 119906)119903minus1

times sum

|120572|1=119903

10038171003817100381710038171198921003817100381710038171003817119901119903infin

120572int

Ω

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

|120572|119889120588 (119905) 119889119906

le 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588 (119905)

(66)

By (14) and the Holder inequality

(intΩ

1003816100381610038161003816119870119899(119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588(119905))

119901(119909)

le intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119901(119909)119889120588 (119905) 119862

119901(119909)minus1

0

(67)

If we denote the largest integer 119903minus satisfying 2119903minus le 119903119901minus as 119903minusand the smallest integer 119903+ satisfying 2119903+ ge 119903119901+ we find

|119905 minus 119909|119903119901(119909)

le |119905 minus 119909|

119903119901+ le |119905 minus 119909|2119903+ if |119905 minus 119909| ge 1

|119905 minus 119909|119903119901minus le |119905 minus 119909|

2119903minus if |119905 minus 119909| lt 1(68)

We combine this with (16) and see that for 120582 = 119889119903119892

119901119903infin

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903+ + |119905 minus 119909|2119903minus 119889120588 (119905) 119889120588 (119909)

le 119862119901+minus1

0(119862119903+

119899minus119903+ + 119862119903minus

119899minus119903minus) le 119862

119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus

(69)

When 119899 ge 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus we take

= 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin(119862119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus)1119901+ (70)

and find

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

)

119901(119909)

119889120588 (119909) le 1 (71)

which implies

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

le le 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

(72)

Thus by Theorem 6 and taking infimum over 119892 isin 119882119903infin

119901 we

have1003817100381710038171003817119871119899(119891) minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

1003817100381710038171003817119871119899(119891 minus 119892)

1003817100381710038171003817119871119901(sdot)120588+

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

+1003817100381710038171003817119892 minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

(119872119901119887 + 1)1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588

+1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot)

(73)

where the constant 119860119901119887119889 is given by

119860119901119887119889 = 119872119901119887 + 1 + 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(74)

When 119899 lt 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus then from the

inequality 119891 minus 119892119871119901(sdot)

120588

+ 119905119892119901119903infin

ge 119905119891119871119901(sdot)

120588

valid for 119905 le 1

we observe

K119903(119891 119899minus119903minus119901+)

119901(sdot)ge 119899

minus119903minus119901+10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

(75)

and applyingTheorem 6 directly yields

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1)1003817100381710038171003817119891

1003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1) 119899119903minus119901+K119903(119891 119899

minus119903minus119901+)119901(sdot)

(76)

and thereby (21) by setting

119860119901119887119889 = (119872119901119887 + 1) 1198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(77)

The proof of Theorem 7 is complete

Abstract and Applied Analysis 9

6 Further Topics and Discussion

Approximation by linear operators is an important topicin approximation theory It mainly consists of two familiesof approximation schemes Bernstein type positive linearoperators and quasi-interpolation type linear operators inmultivariate approximation

In this paper we mainly consider Bernstein type pos-itive linear operators We verified a conjecture in [11]about the uniformboundedness of Bernstein-Durrmeyer andBernstein-Kantorovich operators with respect to an arbitraryBorel measure on (0 1) on the variable 119871

119901(sdot)

120588space under

the assumption of log-Holder continuity of the exponentfunction 119901 We also provide quantitative estimates for highorders of approximation on the variable 119871

119901(sdot)

120588by linear

combinations of Bernstein type positive linear operatorsThe study of quasi-interpolation type linear operators

started with the classical work of Schoenberg on cardinalinterpolation by B-splines It has been developed significantlydue to important applications in the areas of finite elementmethods cardinal interpolation for multivariate approxima-tion and wavelet analysis A large class of linear operators forapproximating functions on R119889 take the form

119879 (119891 119909) = intR119889

Φ (119909 119905) 119891 (119905) 119889119905 119909 isin R119889 (78)

where Φ R119889times R119889

rarr R is a window functionsatisfying int

R119889Φ(119909 119905)119889119905 = 1 and some conditions for decays

of |Φ(119909 119905) as |119909 minus 119905| increases Quantitative estimates for theapproximation of functions in 119862(R119889

) cap 119871infin

(R119889) or 119871

119901(R119889

)

with 1 le 119901 lt infin can be found in a large literature ofmultivariate approximation (see eg [20 21 28]) Establish-ing analysis for approximation by quasi-interpolation typelinear operators on the variable 119871

119901(sdot)

120588spaces would be an

interesting topic An immediate barrier we meet with suchanalysis is the boundedness assumption of the measure 120588

(120588(Ω) lt infin) This assumption is not satisfied for mostquasi-interpolation type linear operators or the classicalWeierstrass (or Gaussian convolution) operators W119899(119891) =

(radic1198992120587)119889

intR119889

expminus119899119905 minus 1199092

22119891(119905)119889119905 for which 120588 is often

the Lebesgue measure on R119889 It is desirable to overcomethe technical difficulty and establish error analysis for linearoperators with respect to unbounded measures

We describedmotivations of our study in learning theoryIt would be interesting to implement detailed error analysisfor some related learning algorithms in classification andquantile regression by means of our results on orders ofapproximation by linear operators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their constructive suggestions and comments The work

described in this paper is supported partially by the ResearchGrants Council of Hong Kong (Project no CityU 105011)Thecorresponding author is Ding-Xuan Zhou

References

[1] S N Bernstein ldquoDemonstration du teoreme de Weirerstrassfondee sur le calcul des probabilitesrdquo Communications of theKharkov Mathematical Society vol 13 pp 1ndash2 1913

[2] L V Kantorovich ldquoSur certaines developments suivant lespolynomes de la forme de S Bernstein I-IIrdquo Comptes Rendus delrsquoAcademie des Sciences de LrsquoURSS A vol 563ndash568 pp 595ndash6001930

[3] J L Durrmeyer Une formule drsquoinversion de la transformeeLaplace applications a la theorie des moments [These de 3e cycleSciences] Faculte des Sciences lrsquoUniversite Paris Paris France1967

[4] H Berens and G G Lorentz ldquoInverse theorems for Bernsteinpolynomialsrdquo Indiana University Mathematics Journal vol 21pp 693ndash708 1972

[5] H Berens and R A DeVore ldquoQuantitative Korovkin theoremsfor positive linear operators on 119871119875-spacesrdquo Transactions of theAmerican Mathematical Society vol 245 pp 349ndash361 1978

[6] Z Ditzian and V TotikModuli of Smoothness vol 9 of SpringerSeries in Computational Mathematics Springer New York NYUSA 1987

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents Springer BerlinGermany 2011

[8] W Orlicz ldquoUber konjugierte Exponentenfolgenrdquo Studia Math-ematica vol 3 pp 200ndash211 1931

[9] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[10] O Kovacik and J Rakosnk ldquoOn spaces 119871119901(119909) and 119882

1119901(119909)rdquoCzechoslovakMathematical Journal vol 41 no 116 pp 592ndash6181991

[11] D X Zhou ldquoApproximation by positive linear operators onvariables 119871

119901(119909) spacesrdquo Journal of Applied Functional Analysisvol 9 no 3-4 pp 379ndash391 2014

[12] D X Zhou and K Jetter ldquoApproximation with polynomialkernels and SVM classifiersrdquo Advances in Computational Math-ematics vol 25 no 1ndash3 pp 323ndash344 2006

[13] E E Berdysheva and K Jetter ldquoMultivariate Bernstein-Durrmeyer operators with arbitrary weight functionsrdquo Journalof Approximation Theory vol 162 no 3 pp 576ndash598 2010

[14] A B Tsybakov ldquoOptimal aggregation of classifiers in statisticallearningrdquo The Annals of Statistics vol 32 no 1 pp 135ndash1662004

[15] S Smale andD X Zhou ldquoLearning theory estimates via integraloperators and their approximationsrdquo Constructive Approxima-tion vol 26 no 2 pp 153ndash172 2007

[16] S Smale and D X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquoThe American MathematicalSociety Bulletin vol 41 no 3 pp 279ndash305 2004

[17] T Hu J Fan Q Wu and D X Zhou ldquoRegularization schemesfor minimum error entropy principlerdquo Analysis and Applica-tions 2014

[18] I Steinwart and A Christmann ldquoEstimating conditional quan-tiles with the help of the pinball lossrdquo Bernoulli vol 17 no 1 pp211ndash225 2011

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

functions119891 onΩ such thatintΩ

(|119891(119909)|120582)119901(119909)

119889120588 lt infin for some120582 gt 0 The norm is defined by

10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

= inf 120582 gt 0 intΩ

(

1003816100381610038161003816119891 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 le 1 (3)

The space 119871119901(sdot)

120588(Ω) is a Banach space [7] The idea of

variable 119871119901(sdot) spaces was introduced by Orlicz [8] Motivated

by connections to variational integrals with nonstandardgrowth related to modeling of electrorheological fluids [9]these function spaces have been developed in analysis andresearch topics include boundedness of maximal operatorscontinuity of translates and denseness of smooth functionsWe will not go into details which can be found in [710] and references therein Instead we only mention thefollowing core condition on the log-Holder continuity of theexponent function which leads to the boundedness of Hardy-Littlewood maximal operators and the rich theory of thevariable 119871

119901(sdot)

120588(Ω) spaces

Definition 1 We say that the exponent function 119901 Ω rarr

[1 infin) is log-Holder continuous if there exist positive con-stants 119860119901 gt 0 such that

1003816100381610038161003816119901 (119909) minus 119901 (119910)1003816100381610038161003816 le

119860119901

minus log 1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119909 119910 isin Ω1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 lt1

2

(4)

We say that 119901 is log-Holder continuous at infinity (when Ω isunbounded) if there holds

1003816100381610038161003816119901 (119909) minus 119901 (119910)1003816100381610038161003816 le

119860119901

log (119890 + |119909|) 119909 119910 isin Ω

10038161003816100381610038161199101003816100381610038161003816 ge |119909| (5)

Denote

119901minus = inf119909isinΩ

119901 (119909) 119901+ = sup119909isinΩ

119901 (119909) (6)

The issue of approximation by Bernstein type positivelinear operators on variable 119871

119901(sdot)

120588(Ω) spaces was raised by the

second author in [11] It turned out that the variety of theexponent function 119901 creates technical difficulty in the studyof approximation In particular the uniform boundednessof the Bernstein-Kantorovich operators (1) and Bernstein-Durrmeyer operators (2) is already a difficult problem Thekey analysis in [11] is to show that the Bernstein-Kantorovichoperators and Bernstein-Durrmeyer operators are uniformlybounded when the exponent function 119901 is Lipschitz 120572 forsome 120572 isin (0 1] It was conjectured there that the uniformboundedness still holds when 119901 is log-Holder continuousThe firstmain result of this paper is to confirm this conjectureinTheorem 6 below

Our second main result is to abandon the positivity andpresent quantitative estimates for the high order approxi-mation by linear operators including linear combinations ofBernstein type positive linear operators extending the resultsin [11] for the first order approximation by positive operators

2 Motivations from Learning Theory

Our main motivation for considering the approximation offunctions by linear operators on variable 119871

119901(sdot)

120588(Ω) spaces is

from learning theory Besides the example of extending theBernstein-Durrmeyer operators (2) to those associated witha general probability measure 120588 on Ω in [12 13] for themultivariate case we mention two learning theory settingshere Since error analysis for concrete learning algorithmsin terms of the introduced noise conditions involves sampleerror estimates which are out of the scope of this paper weleave the detailed error bounds to our further study

21 Noise Conditions for Classification and ApproximationThe first learning theory setting related to approximationon variable 119871

119901(sdot)

120588(Ω) spaces is noise conditions for binary

classification Here Ω is an input space consisting of possibleevents while the output space is denoted as 119884 = 1 minus1A Borel probability measure 119875 on the product space Ω times 119884

can be decomposed into its marginal distribution 119875Ω on Ω

and conditional distributions 119875(sdot | 119909) for 119909 isin Ω A binaryclassifier 119891 Ω rarr 119884 makes predictions 119891(119909) isin 119884 for futureevents 119909 isin Ω The best classifier 119891119888 called Bayes rule is givenby 119891119888(119909) = 1 if 119875(1 | 119909) gt 12 and minus1 otherwise Theprobability measure 119875 fits the binary classification problemwell if the conditional probabilities 119875(1 | 119909) and 119875(minus1 | 119909)

are well separated from the boundary 12 for most events119909 Their separations are equivalent to the separation of thevalue 119891119875(119909) of the regression function 119891119875(119909) = int

119884119910119889119875(119910 |

119909) = 119875(1 | 119909) minus 119875(minus1 | 119909) from 0 and can be measured invarious quantitative ways The Tsybakov noise condition [14]with noise exponent 119902 isin (0 infin] asserts that for some constant119888119902 gt 0 there holds

119875Ω (119909 isin Ω 0 lt1003816100381610038161003816119891119875 (119909)

1003816100381610038161003816 le 119888119902119905) le 119905119902 forall119905 gt 0 (7)

When 119902 = infin Tsybakov noise condition (7) means |119891119875(119909)| ge

119888119902 almost surely and 119891119875(119909) is well separated from 0 Thecase 119902 lt infin means the measure of the set of events 119909

with 119891119875(119909) not well separated from 0 decays polynomiallyfast as the threshold 119888119902119905 tends to 0 More details about theTsybakov noise condition the so-called Tsybakov functionand its applications to the study of classification problems canalso be found in [15] Here we introduce a noise conditionby allowing some noise situations measured by an exponentfunction 119901

Example 2 We say that the probability measure 119875 satisfiesthe noise condition associated with an exponent function119901 Ω rarr [0 infin) if for some 120582 gt 0 there holds

intΩ

(

1003816100381610038161003816119891119875 (119909)1003816100381610038161003816

120582)

119901(119909)

119889119875Ω lt infin (8)

Remark 3 The above condition can be applied to the regres-sion setting for dealingwith unbounded regression functionsWhen 119901 takes values on [1 infin) the above condition isequivalent to the requirement 119891119875 isin 119871

119901(sdot)

120588(Ω) with 120588 = 119875Ω

Abstract and Applied Analysis 3

When 119889 = 1 and Ω = R we apply the classical identityintR

119892(119909)119889119875Ω = intinfin

0119875Ω(119909 isin Ω 119892(119909) ge 119905)119889119905 to the

nonnegative function 119892(119909) = (|119891119875(119909)|120582)119901(119909) and find that the

above condition is equivalent to

int

infin

0

119875Ω (119909 isin Ω 1003816100381610038161003816119891119875 (119909)

1003816100381610038161003816 ge 1205821199051119901(119909)

) 119889119905 lt infin (9)

This illustrates some similarity between the noise condition(8) and Tsybakov noise condition (7)

The following is an example to show some differences

Example 4 Let 119889 isin N and Ω = 119909 isin R119889 |119909| le 12 If 119875Ω is

the normalized Lebesgue measure on Ω and 119891119875(119909) = 119890minus1|119909|

then the measure 119875 satisfies the noise condition associatedwith the exponent function119901(119909) = |119909| but does not satisfy theTsybakov noise condition (7) with any 119902 isin (0 infin] In fact wehave int

Ω|119891119875(119909)|

119901(119909)119889119875Ω = 1119890 lt infin while for any 119902 isin (0 infin)

and 119888119902 gt 0 we have 119875Ω(119909 isin Ω 0 lt |119891119875(119909)| le 119888119902119905) =

(2 minus log(119888119902119905))119889

gt 119905119902 for 119905 isin (0min119905

lowast 1119888119902119890

2) with 119905

lowast beingthe positive solution to the equation 119905

119902119889 log 1119888119902119905 = 2

22 Noise Conditions for Quantile Regression and Approxima-tion The second learning theory setting related to approx-imation on variable 119871

119901(sdot)

120588(Ω) spaces is noise conditions for

quantile regression Here the output space is 119884 = R Similarto the least squares regression [16] for learning means ofconditional distributions 119875(sdot | 119909) but providing richerinformation [17] about response variables such as stretchingor compressing tails the learning problem for quantile regres-sion aims at estimating quantiles of conditional distributionsWith a quantile parameter 0 lt 120591 lt 1 the value of a quantileregression function119891119875120591 at119909 isin Ω is defined by its value119891119875120591(119909)

as a 120591-quantile of 119875(sdot | 119909) that is a value 119905lowast

isin 119884 satisfying

119875 (119910 isin 119884 119910 le 119905lowast | 119909) ge 120591

119875 (119910 isin 119884 119910 ge 119905lowast | 119909) ge 1 minus 120591

(10)

Quantile regression has been studied by kernel-based regu-larization schemes in a learning theory literature (eg [1819]) For optimal error analysis of these learning algorithmsasymptotic behaviors of the conditional distributions near the120591-quantiles are needed In particular one is interested in howslow the following function decays as 119905 decreases

119865119875120591 (119909) = min 119875 (119910 isin 119884 119905lowast

minus 119905 lt 119910 lt 119905lowast | 119909)

119875 (119910 isin 119884 119905lowast

+ 119905 gt 119910 gt 119905lowast | 119909)

(11)

A noise condition was introduced in [18] by requiring lowerbounds 119865119875120591(119909) ge 119887119909119905

119902minus1 for every 119905 isin [0 119886119909] and some119902 isin (1 infin) 119901 isin (0 infin) and constants 119887119909 119886119909 gt 0

satisfying (119887119909119886119902minus1

119909)minus1

isin 119871119901

119875Ω This condition was extended to

a logarithmic bound in [19] by replacing 119905119902minus1 by (log(1119905))

minus119902

and 119886119902minus1

119909by (log(1119886119909))

minus119902 Here we introduce the followingnoise condition which is more general than the one in [18] byallowing the indices 119902 119901 to depend on the events 119909 isin Ω

Example 5 We say that the probabilitymeasure119875 satisfies thequantile noise condition associated with exponent functions119902 Ω rarr (1 infin) and 119901 Ω rarr (0 infin) if for every 119909 isin 119883there exist a 120591-quantile 119905

lowastisin R and constants 119886119909 isin (0 2] 119887119909 gt

0 such that for each 119906 isin [0 119886119909]

119865119875120591 (119909) ge 119887119909119905119902(119909)minus1 (12)

and that for some 120582 gt 0 there holdsintΩ

(1120582119887119909119886119902(119909)minus1

119909)119901(119909)

119889119875Ω lt infin

While the lower bounds (12) imply polynomial decaysof the conditional distributions near the 120591-quantiles with apower index depending on the event the finiteness of theintegral is equivalent to the requirement that the function1119887119909119886

119902(119909)minus1

119909lies in the variable 119871

119901(sdot)

119875Ω(Ω) space (when 119901 takes

values in [1 infin))

3 Main Results for Approximation on 119871119901(sdot)

120588

Our first theorem is about the uniform boundedness of asequence of linear operators on the variable119871

119901(sdot)

120588spacesThese

operators take the form

119871119899 (119891 119909) = intΩ

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905) 119909 isin Ω 119891 isin 119871119901(sdot)

120588 (13)

in terms of their kernels 119870119899(119909 119905)infin

119899=1defined on Ω times Ω We

assume that the kernels satisfy the following three conditionswith some positive constants 1198620 ge 1 119887 119862119887 and 119862119903

(depending on 119903 isin N)

sup119905isinΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119909) le 1198620

sup119909isinΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) le 1198620

(14)

sup119909119905isinΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 le 119862119887119899

119887 forall119899 isin N (15)

intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909) le 119862119903119899

minus119903

forall119899 isin N 119903 isin N

(16)

Then the uniform boundedness follows which will beproved in Section 5

Theorem 6 Let Ω sube R119889 be an open set and an exponentfunction 119901 Ω rarr (1 infin) satisfy 1 lt 119901minus lt 119901+ lt infin and thelog-Holder continuity condition (4) If the kernels 119870119899(119909 119905)

infin

119899=1

satisfy conditions (14) (15) and (16) then the operators 119871119899infin

119899=1

on 119871119901(sdot)

120588defined by (13) are uniformly bounded as

1003817100381710038171003817119871119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (17)

by a positive constant 119872119901119887 (depending on 119901 and the constantsin (14) (15) and (16) given explicitly in the proof)

4 Abstract and Applied Analysis

Our second theorem gives orders of approximation whenthe approximated function has some smoothness stated interms of a K-functional Define a Holder space with index119903 isin N on Ω by

119882119903infin

119901= 119892 isin 119871

119901(sdot)

1205881003817100381710038171003817119892

1003817100381710038171003817119901119903infin lt infin (18)

where 119892119901119903infin

is the norm given by 119892119901119903infin

= 119892119871119901(sdot)

120588

+

sum|120572|1le119903119863

120572119892

infinwith 119863

120572119892 = 120597

|120572|112059711990912057211

sdot sdot sdot 120597119909120572119889

119889and |120572|1 =

sum119889

119895=1120572119895 for 120572 = (1205721 120572119889) isin Z119889

+ The K-functional

K119903(119891 119905)119901(sdot) is defined by

K119903(119891 119905)119901(sdot)

= inf119892isin119882119903infin119901

1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588+ 119905

10038171003817100381710038171198921003817100381710038171003817119901119903infin 119905 gt 0

(19)

Denote 119862infin

0(Ω) as the space of all compactly supported

119862infin functions on Ω From [7] we know that when 119901

+lt

infin 119862infin

0(Ω) is dense in 119871

119901(sdot)

120588(Ω) Hence for any 119891 isin 119871

119901(sdot)

120588(Ω)

there holdsK119903(119891 119905)119901(sdot) rarr 0 as 119905 rarr 0The following theorem to be proved in Section 5 and

extending the results for 119903 = 1 in [11] gives orders ofapproximation by linear operators on 119871

119901(sdot)

120588(Ω) when the K-

functional has explicit decay rates

Theorem 7 Under the assumption of Theorem 6 if Ω isconvex 119903 isin N and the kernels satisfy 119903119901minus ge 2 and

intΩ

119870119899 (119909 119905) (119905 minus 119909)120572119889120588 (119905) = 1205751205720 forall120572 isin Z

119889119908119894119905ℎ |120572|1 lt 119903

(20)

for almost every 119909 isin Ω then there holds for any 119891 isin 119871119901(sdot)

120588

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) (21)

where 119903minus is the integer part of 119903119901minus2 and the constant 119860119901119887119889 isindependent of 119891 isin 119871

119901(sdot)

120588(given explicitly in the proof)

The vanishing moment assumption (20) corresponds toStrang-Fix type conditions in the literature of shift-invariantspaces for example [20 21] It has appeared in the literatureof Bernstein type operators when linear combinations areconsidered as described by (34) in the next section

4 Approximation by Bernstein Type Operators

In this section we apply our main results to Bernstein typepositive linear operators and give high orders of approxima-tion by linear combinations of these operators on variable119871119901(sdot)

120588(Ω) spaces We demonstrate the analysis for the general

Bernstein-Durrmeyer operators in detail and describe brieflyresults for the general Bernstein-Kantorovich operators as anexample of other families of operators

The Bernstein-Durrmeyer operators on an open simplex

Ω = 119878 = 119909 isin R119889

+ |119909|1 lt 1 (22)

associated with a general positive Borel measure 120588 on 119878 aredefined as

119863119899 (119891 119909) = sum

|120572|1le119899

int119878

119891 (119905) 119901119899120572 (119905) 119889120588 (119905)

int119878

119901119899120572 (119905) 119889120588 (119905)119901119899120572 (119909)

119891 isin 1198711

120588 119909 isin 119878

(23)

where for 119909 = (1199091 119909119889) isin 119878 sub R119889 120572 = (1205721 120572119889) isin Z119889

+

we denote

119901119899120572 (119909) =119899

Π119889119895=1

120572119895 (119899 minus |120572|1)

119889

prod

119895=1

119909120572119895

119895(1 minus |119909|1)

119899minus|120572|1 (24)

The classical Bernstein-Durrmeyer operators (2) on Ω =

(0 1) (119889 = 1) with 119889120588(119909) = 119889119909 have been well studied (eg[22]) and extended to a multivariate form with respect toJacobi weights 119889120588(119909) = Π

119889

119895=1119909119902119895

119895119889119909 (eg [23]) Bernstein-

Durrmeyer operators on Ω = (0 1) with respect to anarbitrary Borel probability measure were introduced in [12]and applied to error analysis of learning algorithms forsupport vectormachine classificationsThemultidimensionalversion of such linear operators (23)was introduced in [13] In[24] the first author showed for a constant exponent function119901(119909) equiv 119901 isin [1 infin) that lim119899rarrinfin119891 minus 119863119899(119891)

119871119901

120588= 0 for any

119891 isin 119871119901

120588 The case 119901 equiv infin was studied in [25 26] Here we

consider the case with a general exponent function satisfying1 lt 119901minus lt 119901+ lt infin and the log-Holder continuity condition(4)

By applyingTheorem 6we can prove the uniformbound-edness of the Bernstein-Durrmeyer operators (23)

Proposition 8 LetΩ = 119878 120588 be a Borel probability measure on119878 and an exponent function 119901 119878 rarr (1 infin) satisfy 1 lt 119901minus lt

119901+ lt infin and the log-Holder continuity condition (4) If thereexist positive constants 119887 and 119862

lowast

119887such that int

119878119901119899120572(119905)119889120588(119905) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Durrmeyer operators defined on 119871119901(sdot)

120588(Ω) by (23) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119863119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (25)

Proof Define a sequence of kernels Ψ119899 on 119878 times 119878 by

Ψ119899 (119909 119905) = sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905) (26)

Then the Bernstein-Durrmeyer operators (23) can be writtenas

119863119899 (119891 119909) = int119878

Ψ119899 (119909 119905) 119891 (119905) 119889120588 (119905) (27)

So we only need to check the three conditions (14) (15) and(16) of Theorem 6

Abstract and Applied Analysis 5

Since Ψ119899(119909 119905) ge 0 we know that

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 119889120588 (119909) = int

119878

Ψ119899 (119909 119905) 119889120588 (119909)

= sum

|120572|1le119899

119901119899120572 (119905) int119878

119901119899120572 (119909) 119889120588 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

= sum

|120572|1le119899

119901119899120572 (119905) equiv 1

(28)

The same is true for int119878

|Ψ119899(119909 119905)|119889120588(119905) equiv 1 So we know thatcondition (14) holds with the constant 1198620 = 1

Applying the lower bound int119878

119901119899120572(119905)119889120588(119905) ge 119862lowast

119887119899minus119887 and

the inequality 119901119899120572(119905) le 1 we see that for any 119899 isin N and119909 119905 isin 119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 le sum

|120572|1le119899

119901119899120572 (119909)

119862lowast

119887119899minus119887

=1

119862lowast

119887

119899119887 (29)

Hence condition (15) holds with 119862119887 = 1119862lowast

119887

As for the last condition we separate 119905 minus 119909 into 119905 minus (120572119899) +

(120572119899) minus 119909 and find that for any 119903 isin N and 119899 isin N

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816|119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909)

le 22119903

int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

+ int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

= 22119903

int119878

sum

|120572|1le119899

119901119899120572 (119905)1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905)

+ int119878

sum

|120572|1le119899

119901119899120572 (119909)1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119909)

= 22119903+1

int119878

119861119899 ((sdot minus 119909)2119903

119909) 119889120588 (119909)

(30)

where 119861119899 is the multidimensional Bernstein operators on theclosure 119878 of 119878 defined by

119861119899 (119891 119909) = sum

|120572|1le119899

119891 (120572

119899) 119901119899120572 (119909) 119909 isin 119878 119891 isin 119862 (119878) (31)

It is well known [6] for the multidimensional Bernsteinoperators that there exists a constant 119862119889119903 depending only on119889 and 119903 such that

119861119899 ((sdot minus 119909)2119903

119909) le 119862119889119903119899minus119903

forall119909 isin 119878 (32)

It follows that

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909) le 2

2119903+1119862119889119903119899

minus119903 (33)

and condition (16) holds true with 119862119903 = 22119903+1

119862119889119903With all the three conditions verified the desired uniform

bound (25) for the Bernstein-Durrmeyer operators followsfromTheorem 6 This proves the proposition

The Bernstein-Durrmeyer operators (23) are positivewhich prevent from achieving high order approximation dueto a saturation phenomenon Linear combinations of suchoperators can be used to get high orders of approximationThe idea and literature review of this method can be found in[6] while further developments will not be mentioned hereThe linear combinations are defined as

119871119899119903 (119891 119909) =

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894(119891 119909) (34)

where 119898119889119903 = (119889 + 119903 minus 1)119889(119903 minus 1) is the dimension of thespace of polynomials of degree at most 119903 minus 1 and with twopositive constants 1198611 1198612 independent of 119899 we have

119899 = 1198990 lt 1198991 lt sdot sdot sdot lt 119899119898119889119903le 1198611119899

119898119889119903

sum

119894=0

1003816100381610038161003816119862119894 (119899)1003816100381610038161003816 le 1198612

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894((sdot minus 119909)

120572 119909) = 1205751205720 forall0 le |120572|1 le 119903 minus 1

(35)

For the classical Bernstein-Durrmeyer operators with respectto the Lebesgue measure (or even the Jacobi weights) theexistence of the above linear combinations can be seenand found in the literature The existence of such linearcombinations with respect to the arbitrary measure 120588 isa nontrivial problem and deserves intensive study Thistechnical question is out of the scope of this paper and willbe discussed in our further work Here we concentrate on thevariable 119871

119901(sdot)

120588(Ω) spaces and state the following result for the

high orders of approximation under the condition (35) whichis an immediate consequence of Theorem 7

Proposition 9 Under the assumption of Proposition 8 if 2 le

119903 isin 119873 and the operators 119871119899119903119899isinN defined by (34) satisfy (35)then for any 119891 isin 119871

119901(sdot)

120588 we have

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) (36)

where 119903minus is the integer part of 119903119901minus2 and the constant 119860119901119887119889 isindependent of 119891 isin 119871

119901(sdot)

120588

Let us now briefly describe approximation results for theBernstein-Kantorovich operators on 119878 defined [27] as

119861119870119899 (119891 119909) = sum

|120572|1le119899

int119878119899120572

119891 (119905) 119889120588 (119905)

120588 (119878119899120572)119901119899120572 (119909) 119891 isin 119871

1

120588 119909 isin 119878

(37)

6 Abstract and Applied Analysis

where 119878119899120572120572 are subdomains of 119878 defined by

119878119899120572 = 119909 isin 119878 119909 isin

119889

prod

119894=1

[120572119894

119899 + 1

120572119894

119899 + 1) |119909|1 le

|120572|1 + 1

119899 + 1

|120572|1 le 119899

(38)

In the same way as for the Bernstein-Durrmeyer oper-ators we have the following results for the Bernstein-Kantorovich operators

Proposition 10 Under the assumption of Proposition 8 for 119901if there exist positive constants 119887 and 119862

lowast

119887such that 120588(119878119899120572) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Kantorovich operators defined on 119871119901(sdot)

120588(Ω) by (37) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119861119870119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (39)

If 2 le 119903 isin 119873 and with 119863119899119894replaced by 119861119870119899119894

the operators119871119899119903119899isinN defined by (34) satisfy (35) then

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) forall119891 isin 119871

119901(sdot)

120588

(40)

5 Proof of Main Results

In this section we give detailed proof of our main results Letus first proveTheorem 6

Proof of Theorem 6 Let 119891 isin 119871119901(sdot) have norm 1 which implies

intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 Choose 120574 = 119887(119901minus minus 1) gt 0For 119909 isin Ω we define two subsets Ω119899119909 and Ω

1015840

119899119909of Ω as

Ω119899119909 = 119905 isin Ω 1003816100381610038161003816119891 (119905)

1003816100381610038161003816 le 119899120574 |119905 minus 119909| le 119899

minus14

Ω1015840

119899119909= 119905 isin Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le 119899

120574 |119905 minus 119909| gt 119899

minus14

(41)

Set

1198711198991 (119891 119909) = intΩ(Ω119899119909cupΩ

1015840119899119909)

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198992 (119891 119909) = intΩ119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198993 (119891 119909) = intΩ1015840119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

(42)

Then the value 119871119899(119891 119909) can be decomposed into three partsas

119871119899 (119891 119909) = 1198711198991 (119891 119909) + 1198711198992 (119891 119909) + 1198711198993 (119891 119909) 119909 isin Ω

(43)

and we have

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 3119901+

3

sum

119895=1

intΩ

10038161003816100381610038161003816119871119899119895 (119891 119909)

10038161003816100381610038161003816

119901(119909)

119889120588 (119909)

(44)

In the followingwe estimate the three terms in (44) separately

Step 1 Estimating the First Term of (44) By the definition of119901minus we have 119901(119905) ge 119901minus gt 1 For 119905 isin Ω (Ω119899119909 cup Ω

1015840

119899119909) we have

|119891(119905)| gt 119899120574 and thereby

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le

1003816100381610038161003816119891 (119905)1003816100381610038161003816 (

1003816100381610038161003816119891 (119905)1003816100381610038161003816 119899

minus120574)119901(119905)minus1

=1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901(119905))

le1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901minus) = 119899

minus1198871003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

(45)

It follows from condition (15) that

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119899

minus119887intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905)

le 119899minus119887

119862119887119899119887

intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

(46)

So by the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119862119887 int

Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905) le 119862119887 (47)

Consequently

intΩ

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 120588 (Ω) (119862119887

119901minus+ 119862119887

119901+) (48)

Step 2 Estimating the Second Term of (44) By the conditionintΩ

|119870119899(119909 119905)|119889120588(119905) le 1198620 in (14) with 1198620 ge 1 we know by theHolder inequality

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)

119889120588 (119905)

times (intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905))

119901(119909)minus1

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

times 119889120588 (119905) 119889120588 (119909)

= 119862119901+minus1

0intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

(49)

where

119869119899 (119891 119909 119905) =1003816100381610038161003816119870119899 (119909 119905)

10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

119905 isin Ω119899119909 119909 isin Ω

(50)

For 119905 isin Ω119899119909 we have a bound |119891(119905)| le 119899120574 and the

restriction |119909 minus 119905| le 119899minus14 From the log-Holder continuity

of the exponent function 119901 there exists a constant 119860119901 onlydependent on 119901 such that

119901 (119909) minus 119901 (119905) le119860119901

minus log 119899minus14=

4119860119901

log 119899 (51)

Abstract and Applied Analysis 7

When |119891(119905)| ge 1 we find

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119909)minus119901(119905)

le (119899120574)4119860119901 log 119899

le 1198994120574119860119901 log 119899 le 119901 (52)

where the constant number 119901 defined by

119901 = sup119899isinN

1198994120574119860119901 log 119899 (53)

is finite because

log 1198994120574119860119901 log 119899 =

4120574119860119901 log 119899

log 119899997888rarr 4120574119860119901 as 119899 997888rarr infin

(54)

When |119891(119905)| lt 1 we simply use |119891(119905)|119901(119909)minus119901(119905)

le

|119891(119905)|minus119901(119905) Applying these bounds we can estimate the core

part of Step 2 as

intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

(119901 +1003816100381610038161003816119891 (119905)

1003816100381610038161003816minus119901(119905)

)

times 119889120588 (119905) 119889120588 (119909)

= 119901 intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

+ intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) 119889120588 (119909)

le 1199011198620 intΩ

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

+ 1198620120588 (Ω) le 1198620119901 + 1198620120588 (Ω)

(55)

Here we have used the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 and(14) So the second term of (44) can be estimated as

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+0

(119901 + 120588 (Ω)) (56)

Step 3 Estimating the Third Term of (44) For 119905 isin Ω1015840

119899119909 we

have |119891(119905)| le 119899120574 and |119909 minus 119905| gt 119899

minus14 yielding 11989914

|119909 minus 119905| gt 1 Itfollows that

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816 le int

Ω1015840119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119899

120574(119899

14|119909 minus 119905|)

2119903119901(119909)

119889120588 (119905)

(57)

where 119903 is the integer part of 2120574119901+ + 1 Applying the Holderinequality and (14) we see that

10038161003816100381610038161198711198993(119891 119909)1003816100381610038161003816119901(119909)

le intΩ1015840119899119909

119899120574119901(119909)+(1199032) 1003816100381610038161003816119870119899 (119909 119905)

1003816100381610038161003816 |119909 minus 119905|2119903

119889120588 (119905) 119862119901(119909)minus1

0

(58)

So by the bound 119901(119909) le 119901+ and condition (16) the third termof (44) can be bounded as

intΩ

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+minus1

0119899120574119901++(1199032)119862119903119899

minus119903

= 119862119901+minus1

0119862119903119899

120574119901+minus(1199032) le 119862119901+minus1

0119862119903

(59)

Finally we put the estimates (48) (56) and (59) into (44)to conclude

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(60)

Take

120582 = 119872119901119887 = 1 + 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(61)

we find

intΩ

(

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le (1

120582)

119901minus

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 1

(62)

This implies 119871119899(119891)119871119901(sdot)

120588

le 119872119901119887 The bound 119872119901119887 isindependent of 119891 So we have 119871119899 le 119872119901119887 The proofTheorem 6 is complete

We are now in a position to proveTheorem 7

Proof of Theorem 7 We follow the standard procedure inapproximation theory and consider the error 119871119899(119892 119909) minus 119892(119909)

for 119892 isin 119882119903infin

119901 Apply the Taylor expansion

119892 (119905) = 119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119909 119905 isin Ω

(63)

where the remainder term 119877119892119903(119909 119905) is given by

119877119892119903 (119909 119905)

= int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572(119905 minus 119909)

120572119889119906

(64)

8 Abstract and Applied Analysis

We see from the vanishing moment condition (20) that

119871119899 (119892 119909) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119889120588 (119905) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572

times (119905 minus 119909)120572119889119906 119889120588 (119905)

(65)

SinceΩ is convex 119909+119906(119905minus119909) isin Ω for any 119906 isin [0 1] 119909 119905 isin ΩSo |119863

120572119892(119909 + 119906(119905 minus 119909))| le 119892

119901119903infinand we have

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

le int

1

0

(1 minus 119906)119903minus1

times sum

|120572|1=119903

10038171003817100381710038171198921003817100381710038171003817119901119903infin

120572int

Ω

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

|120572|119889120588 (119905) 119889119906

le 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588 (119905)

(66)

By (14) and the Holder inequality

(intΩ

1003816100381610038161003816119870119899(119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588(119905))

119901(119909)

le intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119901(119909)119889120588 (119905) 119862

119901(119909)minus1

0

(67)

If we denote the largest integer 119903minus satisfying 2119903minus le 119903119901minus as 119903minusand the smallest integer 119903+ satisfying 2119903+ ge 119903119901+ we find

|119905 minus 119909|119903119901(119909)

le |119905 minus 119909|

119903119901+ le |119905 minus 119909|2119903+ if |119905 minus 119909| ge 1

|119905 minus 119909|119903119901minus le |119905 minus 119909|

2119903minus if |119905 minus 119909| lt 1(68)

We combine this with (16) and see that for 120582 = 119889119903119892

119901119903infin

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903+ + |119905 minus 119909|2119903minus 119889120588 (119905) 119889120588 (119909)

le 119862119901+minus1

0(119862119903+

119899minus119903+ + 119862119903minus

119899minus119903minus) le 119862

119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus

(69)

When 119899 ge 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus we take

= 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin(119862119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus)1119901+ (70)

and find

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

)

119901(119909)

119889120588 (119909) le 1 (71)

which implies

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

le le 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

(72)

Thus by Theorem 6 and taking infimum over 119892 isin 119882119903infin

119901 we

have1003817100381710038171003817119871119899(119891) minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

1003817100381710038171003817119871119899(119891 minus 119892)

1003817100381710038171003817119871119901(sdot)120588+

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

+1003817100381710038171003817119892 minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

(119872119901119887 + 1)1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588

+1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot)

(73)

where the constant 119860119901119887119889 is given by

119860119901119887119889 = 119872119901119887 + 1 + 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(74)

When 119899 lt 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus then from the

inequality 119891 minus 119892119871119901(sdot)

120588

+ 119905119892119901119903infin

ge 119905119891119871119901(sdot)

120588

valid for 119905 le 1

we observe

K119903(119891 119899minus119903minus119901+)

119901(sdot)ge 119899

minus119903minus119901+10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

(75)

and applyingTheorem 6 directly yields

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1)1003817100381710038171003817119891

1003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1) 119899119903minus119901+K119903(119891 119899

minus119903minus119901+)119901(sdot)

(76)

and thereby (21) by setting

119860119901119887119889 = (119872119901119887 + 1) 1198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(77)

The proof of Theorem 7 is complete

Abstract and Applied Analysis 9

6 Further Topics and Discussion

Approximation by linear operators is an important topicin approximation theory It mainly consists of two familiesof approximation schemes Bernstein type positive linearoperators and quasi-interpolation type linear operators inmultivariate approximation

In this paper we mainly consider Bernstein type pos-itive linear operators We verified a conjecture in [11]about the uniformboundedness of Bernstein-Durrmeyer andBernstein-Kantorovich operators with respect to an arbitraryBorel measure on (0 1) on the variable 119871

119901(sdot)

120588space under

the assumption of log-Holder continuity of the exponentfunction 119901 We also provide quantitative estimates for highorders of approximation on the variable 119871

119901(sdot)

120588by linear

combinations of Bernstein type positive linear operatorsThe study of quasi-interpolation type linear operators

started with the classical work of Schoenberg on cardinalinterpolation by B-splines It has been developed significantlydue to important applications in the areas of finite elementmethods cardinal interpolation for multivariate approxima-tion and wavelet analysis A large class of linear operators forapproximating functions on R119889 take the form

119879 (119891 119909) = intR119889

Φ (119909 119905) 119891 (119905) 119889119905 119909 isin R119889 (78)

where Φ R119889times R119889

rarr R is a window functionsatisfying int

R119889Φ(119909 119905)119889119905 = 1 and some conditions for decays

of |Φ(119909 119905) as |119909 minus 119905| increases Quantitative estimates for theapproximation of functions in 119862(R119889

) cap 119871infin

(R119889) or 119871

119901(R119889

)

with 1 le 119901 lt infin can be found in a large literature ofmultivariate approximation (see eg [20 21 28]) Establish-ing analysis for approximation by quasi-interpolation typelinear operators on the variable 119871

119901(sdot)

120588spaces would be an

interesting topic An immediate barrier we meet with suchanalysis is the boundedness assumption of the measure 120588

(120588(Ω) lt infin) This assumption is not satisfied for mostquasi-interpolation type linear operators or the classicalWeierstrass (or Gaussian convolution) operators W119899(119891) =

(radic1198992120587)119889

intR119889

expminus119899119905 minus 1199092

22119891(119905)119889119905 for which 120588 is often

the Lebesgue measure on R119889 It is desirable to overcomethe technical difficulty and establish error analysis for linearoperators with respect to unbounded measures

We describedmotivations of our study in learning theoryIt would be interesting to implement detailed error analysisfor some related learning algorithms in classification andquantile regression by means of our results on orders ofapproximation by linear operators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their constructive suggestions and comments The work

described in this paper is supported partially by the ResearchGrants Council of Hong Kong (Project no CityU 105011)Thecorresponding author is Ding-Xuan Zhou

References

[1] S N Bernstein ldquoDemonstration du teoreme de Weirerstrassfondee sur le calcul des probabilitesrdquo Communications of theKharkov Mathematical Society vol 13 pp 1ndash2 1913

[2] L V Kantorovich ldquoSur certaines developments suivant lespolynomes de la forme de S Bernstein I-IIrdquo Comptes Rendus delrsquoAcademie des Sciences de LrsquoURSS A vol 563ndash568 pp 595ndash6001930

[3] J L Durrmeyer Une formule drsquoinversion de la transformeeLaplace applications a la theorie des moments [These de 3e cycleSciences] Faculte des Sciences lrsquoUniversite Paris Paris France1967

[4] H Berens and G G Lorentz ldquoInverse theorems for Bernsteinpolynomialsrdquo Indiana University Mathematics Journal vol 21pp 693ndash708 1972

[5] H Berens and R A DeVore ldquoQuantitative Korovkin theoremsfor positive linear operators on 119871119875-spacesrdquo Transactions of theAmerican Mathematical Society vol 245 pp 349ndash361 1978

[6] Z Ditzian and V TotikModuli of Smoothness vol 9 of SpringerSeries in Computational Mathematics Springer New York NYUSA 1987

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents Springer BerlinGermany 2011

[8] W Orlicz ldquoUber konjugierte Exponentenfolgenrdquo Studia Math-ematica vol 3 pp 200ndash211 1931

[9] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[10] O Kovacik and J Rakosnk ldquoOn spaces 119871119901(119909) and 119882

1119901(119909)rdquoCzechoslovakMathematical Journal vol 41 no 116 pp 592ndash6181991

[11] D X Zhou ldquoApproximation by positive linear operators onvariables 119871

119901(119909) spacesrdquo Journal of Applied Functional Analysisvol 9 no 3-4 pp 379ndash391 2014

[12] D X Zhou and K Jetter ldquoApproximation with polynomialkernels and SVM classifiersrdquo Advances in Computational Math-ematics vol 25 no 1ndash3 pp 323ndash344 2006

[13] E E Berdysheva and K Jetter ldquoMultivariate Bernstein-Durrmeyer operators with arbitrary weight functionsrdquo Journalof Approximation Theory vol 162 no 3 pp 576ndash598 2010

[14] A B Tsybakov ldquoOptimal aggregation of classifiers in statisticallearningrdquo The Annals of Statistics vol 32 no 1 pp 135ndash1662004

[15] S Smale andD X Zhou ldquoLearning theory estimates via integraloperators and their approximationsrdquo Constructive Approxima-tion vol 26 no 2 pp 153ndash172 2007

[16] S Smale and D X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquoThe American MathematicalSociety Bulletin vol 41 no 3 pp 279ndash305 2004

[17] T Hu J Fan Q Wu and D X Zhou ldquoRegularization schemesfor minimum error entropy principlerdquo Analysis and Applica-tions 2014

[18] I Steinwart and A Christmann ldquoEstimating conditional quan-tiles with the help of the pinball lossrdquo Bernoulli vol 17 no 1 pp211ndash225 2011

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

When 119889 = 1 and Ω = R we apply the classical identityintR

119892(119909)119889119875Ω = intinfin

0119875Ω(119909 isin Ω 119892(119909) ge 119905)119889119905 to the

nonnegative function 119892(119909) = (|119891119875(119909)|120582)119901(119909) and find that the

above condition is equivalent to

int

infin

0

119875Ω (119909 isin Ω 1003816100381610038161003816119891119875 (119909)

1003816100381610038161003816 ge 1205821199051119901(119909)

) 119889119905 lt infin (9)

This illustrates some similarity between the noise condition(8) and Tsybakov noise condition (7)

The following is an example to show some differences

Example 4 Let 119889 isin N and Ω = 119909 isin R119889 |119909| le 12 If 119875Ω is

the normalized Lebesgue measure on Ω and 119891119875(119909) = 119890minus1|119909|

then the measure 119875 satisfies the noise condition associatedwith the exponent function119901(119909) = |119909| but does not satisfy theTsybakov noise condition (7) with any 119902 isin (0 infin] In fact wehave int

Ω|119891119875(119909)|

119901(119909)119889119875Ω = 1119890 lt infin while for any 119902 isin (0 infin)

and 119888119902 gt 0 we have 119875Ω(119909 isin Ω 0 lt |119891119875(119909)| le 119888119902119905) =

(2 minus log(119888119902119905))119889

gt 119905119902 for 119905 isin (0min119905

lowast 1119888119902119890

2) with 119905

lowast beingthe positive solution to the equation 119905

119902119889 log 1119888119902119905 = 2

22 Noise Conditions for Quantile Regression and Approxima-tion The second learning theory setting related to approx-imation on variable 119871

119901(sdot)

120588(Ω) spaces is noise conditions for

quantile regression Here the output space is 119884 = R Similarto the least squares regression [16] for learning means ofconditional distributions 119875(sdot | 119909) but providing richerinformation [17] about response variables such as stretchingor compressing tails the learning problem for quantile regres-sion aims at estimating quantiles of conditional distributionsWith a quantile parameter 0 lt 120591 lt 1 the value of a quantileregression function119891119875120591 at119909 isin Ω is defined by its value119891119875120591(119909)

as a 120591-quantile of 119875(sdot | 119909) that is a value 119905lowast

isin 119884 satisfying

119875 (119910 isin 119884 119910 le 119905lowast | 119909) ge 120591

119875 (119910 isin 119884 119910 ge 119905lowast | 119909) ge 1 minus 120591

(10)

Quantile regression has been studied by kernel-based regu-larization schemes in a learning theory literature (eg [1819]) For optimal error analysis of these learning algorithmsasymptotic behaviors of the conditional distributions near the120591-quantiles are needed In particular one is interested in howslow the following function decays as 119905 decreases

119865119875120591 (119909) = min 119875 (119910 isin 119884 119905lowast

minus 119905 lt 119910 lt 119905lowast | 119909)

119875 (119910 isin 119884 119905lowast

+ 119905 gt 119910 gt 119905lowast | 119909)

(11)

A noise condition was introduced in [18] by requiring lowerbounds 119865119875120591(119909) ge 119887119909119905

119902minus1 for every 119905 isin [0 119886119909] and some119902 isin (1 infin) 119901 isin (0 infin) and constants 119887119909 119886119909 gt 0

satisfying (119887119909119886119902minus1

119909)minus1

isin 119871119901

119875Ω This condition was extended to

a logarithmic bound in [19] by replacing 119905119902minus1 by (log(1119905))

minus119902

and 119886119902minus1

119909by (log(1119886119909))

minus119902 Here we introduce the followingnoise condition which is more general than the one in [18] byallowing the indices 119902 119901 to depend on the events 119909 isin Ω

Example 5 We say that the probabilitymeasure119875 satisfies thequantile noise condition associated with exponent functions119902 Ω rarr (1 infin) and 119901 Ω rarr (0 infin) if for every 119909 isin 119883there exist a 120591-quantile 119905

lowastisin R and constants 119886119909 isin (0 2] 119887119909 gt

0 such that for each 119906 isin [0 119886119909]

119865119875120591 (119909) ge 119887119909119905119902(119909)minus1 (12)

and that for some 120582 gt 0 there holdsintΩ

(1120582119887119909119886119902(119909)minus1

119909)119901(119909)

119889119875Ω lt infin

While the lower bounds (12) imply polynomial decaysof the conditional distributions near the 120591-quantiles with apower index depending on the event the finiteness of theintegral is equivalent to the requirement that the function1119887119909119886

119902(119909)minus1

119909lies in the variable 119871

119901(sdot)

119875Ω(Ω) space (when 119901 takes

values in [1 infin))

3 Main Results for Approximation on 119871119901(sdot)

120588

Our first theorem is about the uniform boundedness of asequence of linear operators on the variable119871

119901(sdot)

120588spacesThese

operators take the form

119871119899 (119891 119909) = intΩ

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905) 119909 isin Ω 119891 isin 119871119901(sdot)

120588 (13)

in terms of their kernels 119870119899(119909 119905)infin

119899=1defined on Ω times Ω We

assume that the kernels satisfy the following three conditionswith some positive constants 1198620 ge 1 119887 119862119887 and 119862119903

(depending on 119903 isin N)

sup119905isinΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119909) le 1198620

sup119909isinΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) le 1198620

(14)

sup119909119905isinΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 le 119862119887119899

119887 forall119899 isin N (15)

intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909) le 119862119903119899

minus119903

forall119899 isin N 119903 isin N

(16)

Then the uniform boundedness follows which will beproved in Section 5

Theorem 6 Let Ω sube R119889 be an open set and an exponentfunction 119901 Ω rarr (1 infin) satisfy 1 lt 119901minus lt 119901+ lt infin and thelog-Holder continuity condition (4) If the kernels 119870119899(119909 119905)

infin

119899=1

satisfy conditions (14) (15) and (16) then the operators 119871119899infin

119899=1

on 119871119901(sdot)

120588defined by (13) are uniformly bounded as

1003817100381710038171003817119871119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (17)

by a positive constant 119872119901119887 (depending on 119901 and the constantsin (14) (15) and (16) given explicitly in the proof)

4 Abstract and Applied Analysis

Our second theorem gives orders of approximation whenthe approximated function has some smoothness stated interms of a K-functional Define a Holder space with index119903 isin N on Ω by

119882119903infin

119901= 119892 isin 119871

119901(sdot)

1205881003817100381710038171003817119892

1003817100381710038171003817119901119903infin lt infin (18)

where 119892119901119903infin

is the norm given by 119892119901119903infin

= 119892119871119901(sdot)

120588

+

sum|120572|1le119903119863

120572119892

infinwith 119863

120572119892 = 120597

|120572|112059711990912057211

sdot sdot sdot 120597119909120572119889

119889and |120572|1 =

sum119889

119895=1120572119895 for 120572 = (1205721 120572119889) isin Z119889

+ The K-functional

K119903(119891 119905)119901(sdot) is defined by

K119903(119891 119905)119901(sdot)

= inf119892isin119882119903infin119901

1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588+ 119905

10038171003817100381710038171198921003817100381710038171003817119901119903infin 119905 gt 0

(19)

Denote 119862infin

0(Ω) as the space of all compactly supported

119862infin functions on Ω From [7] we know that when 119901

+lt

infin 119862infin

0(Ω) is dense in 119871

119901(sdot)

120588(Ω) Hence for any 119891 isin 119871

119901(sdot)

120588(Ω)

there holdsK119903(119891 119905)119901(sdot) rarr 0 as 119905 rarr 0The following theorem to be proved in Section 5 and

extending the results for 119903 = 1 in [11] gives orders ofapproximation by linear operators on 119871

119901(sdot)

120588(Ω) when the K-

functional has explicit decay rates

Theorem 7 Under the assumption of Theorem 6 if Ω isconvex 119903 isin N and the kernels satisfy 119903119901minus ge 2 and

intΩ

119870119899 (119909 119905) (119905 minus 119909)120572119889120588 (119905) = 1205751205720 forall120572 isin Z

119889119908119894119905ℎ |120572|1 lt 119903

(20)

for almost every 119909 isin Ω then there holds for any 119891 isin 119871119901(sdot)

120588

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) (21)

where 119903minus is the integer part of 119903119901minus2 and the constant 119860119901119887119889 isindependent of 119891 isin 119871

119901(sdot)

120588(given explicitly in the proof)

The vanishing moment assumption (20) corresponds toStrang-Fix type conditions in the literature of shift-invariantspaces for example [20 21] It has appeared in the literatureof Bernstein type operators when linear combinations areconsidered as described by (34) in the next section

4 Approximation by Bernstein Type Operators

In this section we apply our main results to Bernstein typepositive linear operators and give high orders of approxima-tion by linear combinations of these operators on variable119871119901(sdot)

120588(Ω) spaces We demonstrate the analysis for the general

Bernstein-Durrmeyer operators in detail and describe brieflyresults for the general Bernstein-Kantorovich operators as anexample of other families of operators

The Bernstein-Durrmeyer operators on an open simplex

Ω = 119878 = 119909 isin R119889

+ |119909|1 lt 1 (22)

associated with a general positive Borel measure 120588 on 119878 aredefined as

119863119899 (119891 119909) = sum

|120572|1le119899

int119878

119891 (119905) 119901119899120572 (119905) 119889120588 (119905)

int119878

119901119899120572 (119905) 119889120588 (119905)119901119899120572 (119909)

119891 isin 1198711

120588 119909 isin 119878

(23)

where for 119909 = (1199091 119909119889) isin 119878 sub R119889 120572 = (1205721 120572119889) isin Z119889

+

we denote

119901119899120572 (119909) =119899

Π119889119895=1

120572119895 (119899 minus |120572|1)

119889

prod

119895=1

119909120572119895

119895(1 minus |119909|1)

119899minus|120572|1 (24)

The classical Bernstein-Durrmeyer operators (2) on Ω =

(0 1) (119889 = 1) with 119889120588(119909) = 119889119909 have been well studied (eg[22]) and extended to a multivariate form with respect toJacobi weights 119889120588(119909) = Π

119889

119895=1119909119902119895

119895119889119909 (eg [23]) Bernstein-

Durrmeyer operators on Ω = (0 1) with respect to anarbitrary Borel probability measure were introduced in [12]and applied to error analysis of learning algorithms forsupport vectormachine classificationsThemultidimensionalversion of such linear operators (23)was introduced in [13] In[24] the first author showed for a constant exponent function119901(119909) equiv 119901 isin [1 infin) that lim119899rarrinfin119891 minus 119863119899(119891)

119871119901

120588= 0 for any

119891 isin 119871119901

120588 The case 119901 equiv infin was studied in [25 26] Here we

consider the case with a general exponent function satisfying1 lt 119901minus lt 119901+ lt infin and the log-Holder continuity condition(4)

By applyingTheorem 6we can prove the uniformbound-edness of the Bernstein-Durrmeyer operators (23)

Proposition 8 LetΩ = 119878 120588 be a Borel probability measure on119878 and an exponent function 119901 119878 rarr (1 infin) satisfy 1 lt 119901minus lt

119901+ lt infin and the log-Holder continuity condition (4) If thereexist positive constants 119887 and 119862

lowast

119887such that int

119878119901119899120572(119905)119889120588(119905) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Durrmeyer operators defined on 119871119901(sdot)

120588(Ω) by (23) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119863119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (25)

Proof Define a sequence of kernels Ψ119899 on 119878 times 119878 by

Ψ119899 (119909 119905) = sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905) (26)

Then the Bernstein-Durrmeyer operators (23) can be writtenas

119863119899 (119891 119909) = int119878

Ψ119899 (119909 119905) 119891 (119905) 119889120588 (119905) (27)

So we only need to check the three conditions (14) (15) and(16) of Theorem 6

Abstract and Applied Analysis 5

Since Ψ119899(119909 119905) ge 0 we know that

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 119889120588 (119909) = int

119878

Ψ119899 (119909 119905) 119889120588 (119909)

= sum

|120572|1le119899

119901119899120572 (119905) int119878

119901119899120572 (119909) 119889120588 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

= sum

|120572|1le119899

119901119899120572 (119905) equiv 1

(28)

The same is true for int119878

|Ψ119899(119909 119905)|119889120588(119905) equiv 1 So we know thatcondition (14) holds with the constant 1198620 = 1

Applying the lower bound int119878

119901119899120572(119905)119889120588(119905) ge 119862lowast

119887119899minus119887 and

the inequality 119901119899120572(119905) le 1 we see that for any 119899 isin N and119909 119905 isin 119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 le sum

|120572|1le119899

119901119899120572 (119909)

119862lowast

119887119899minus119887

=1

119862lowast

119887

119899119887 (29)

Hence condition (15) holds with 119862119887 = 1119862lowast

119887

As for the last condition we separate 119905 minus 119909 into 119905 minus (120572119899) +

(120572119899) minus 119909 and find that for any 119903 isin N and 119899 isin N

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816|119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909)

le 22119903

int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

+ int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

= 22119903

int119878

sum

|120572|1le119899

119901119899120572 (119905)1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905)

+ int119878

sum

|120572|1le119899

119901119899120572 (119909)1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119909)

= 22119903+1

int119878

119861119899 ((sdot minus 119909)2119903

119909) 119889120588 (119909)

(30)

where 119861119899 is the multidimensional Bernstein operators on theclosure 119878 of 119878 defined by

119861119899 (119891 119909) = sum

|120572|1le119899

119891 (120572

119899) 119901119899120572 (119909) 119909 isin 119878 119891 isin 119862 (119878) (31)

It is well known [6] for the multidimensional Bernsteinoperators that there exists a constant 119862119889119903 depending only on119889 and 119903 such that

119861119899 ((sdot minus 119909)2119903

119909) le 119862119889119903119899minus119903

forall119909 isin 119878 (32)

It follows that

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909) le 2

2119903+1119862119889119903119899

minus119903 (33)

and condition (16) holds true with 119862119903 = 22119903+1

119862119889119903With all the three conditions verified the desired uniform

bound (25) for the Bernstein-Durrmeyer operators followsfromTheorem 6 This proves the proposition

The Bernstein-Durrmeyer operators (23) are positivewhich prevent from achieving high order approximation dueto a saturation phenomenon Linear combinations of suchoperators can be used to get high orders of approximationThe idea and literature review of this method can be found in[6] while further developments will not be mentioned hereThe linear combinations are defined as

119871119899119903 (119891 119909) =

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894(119891 119909) (34)

where 119898119889119903 = (119889 + 119903 minus 1)119889(119903 minus 1) is the dimension of thespace of polynomials of degree at most 119903 minus 1 and with twopositive constants 1198611 1198612 independent of 119899 we have

119899 = 1198990 lt 1198991 lt sdot sdot sdot lt 119899119898119889119903le 1198611119899

119898119889119903

sum

119894=0

1003816100381610038161003816119862119894 (119899)1003816100381610038161003816 le 1198612

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894((sdot minus 119909)

120572 119909) = 1205751205720 forall0 le |120572|1 le 119903 minus 1

(35)

For the classical Bernstein-Durrmeyer operators with respectto the Lebesgue measure (or even the Jacobi weights) theexistence of the above linear combinations can be seenand found in the literature The existence of such linearcombinations with respect to the arbitrary measure 120588 isa nontrivial problem and deserves intensive study Thistechnical question is out of the scope of this paper and willbe discussed in our further work Here we concentrate on thevariable 119871

119901(sdot)

120588(Ω) spaces and state the following result for the

high orders of approximation under the condition (35) whichis an immediate consequence of Theorem 7

Proposition 9 Under the assumption of Proposition 8 if 2 le

119903 isin 119873 and the operators 119871119899119903119899isinN defined by (34) satisfy (35)then for any 119891 isin 119871

119901(sdot)

120588 we have

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) (36)

where 119903minus is the integer part of 119903119901minus2 and the constant 119860119901119887119889 isindependent of 119891 isin 119871

119901(sdot)

120588

Let us now briefly describe approximation results for theBernstein-Kantorovich operators on 119878 defined [27] as

119861119870119899 (119891 119909) = sum

|120572|1le119899

int119878119899120572

119891 (119905) 119889120588 (119905)

120588 (119878119899120572)119901119899120572 (119909) 119891 isin 119871

1

120588 119909 isin 119878

(37)

6 Abstract and Applied Analysis

where 119878119899120572120572 are subdomains of 119878 defined by

119878119899120572 = 119909 isin 119878 119909 isin

119889

prod

119894=1

[120572119894

119899 + 1

120572119894

119899 + 1) |119909|1 le

|120572|1 + 1

119899 + 1

|120572|1 le 119899

(38)

In the same way as for the Bernstein-Durrmeyer oper-ators we have the following results for the Bernstein-Kantorovich operators

Proposition 10 Under the assumption of Proposition 8 for 119901if there exist positive constants 119887 and 119862

lowast

119887such that 120588(119878119899120572) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Kantorovich operators defined on 119871119901(sdot)

120588(Ω) by (37) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119861119870119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (39)

If 2 le 119903 isin 119873 and with 119863119899119894replaced by 119861119870119899119894

the operators119871119899119903119899isinN defined by (34) satisfy (35) then

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) forall119891 isin 119871

119901(sdot)

120588

(40)

5 Proof of Main Results

In this section we give detailed proof of our main results Letus first proveTheorem 6

Proof of Theorem 6 Let 119891 isin 119871119901(sdot) have norm 1 which implies

intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 Choose 120574 = 119887(119901minus minus 1) gt 0For 119909 isin Ω we define two subsets Ω119899119909 and Ω

1015840

119899119909of Ω as

Ω119899119909 = 119905 isin Ω 1003816100381610038161003816119891 (119905)

1003816100381610038161003816 le 119899120574 |119905 minus 119909| le 119899

minus14

Ω1015840

119899119909= 119905 isin Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le 119899

120574 |119905 minus 119909| gt 119899

minus14

(41)

Set

1198711198991 (119891 119909) = intΩ(Ω119899119909cupΩ

1015840119899119909)

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198992 (119891 119909) = intΩ119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198993 (119891 119909) = intΩ1015840119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

(42)

Then the value 119871119899(119891 119909) can be decomposed into three partsas

119871119899 (119891 119909) = 1198711198991 (119891 119909) + 1198711198992 (119891 119909) + 1198711198993 (119891 119909) 119909 isin Ω

(43)

and we have

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 3119901+

3

sum

119895=1

intΩ

10038161003816100381610038161003816119871119899119895 (119891 119909)

10038161003816100381610038161003816

119901(119909)

119889120588 (119909)

(44)

In the followingwe estimate the three terms in (44) separately

Step 1 Estimating the First Term of (44) By the definition of119901minus we have 119901(119905) ge 119901minus gt 1 For 119905 isin Ω (Ω119899119909 cup Ω

1015840

119899119909) we have

|119891(119905)| gt 119899120574 and thereby

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le

1003816100381610038161003816119891 (119905)1003816100381610038161003816 (

1003816100381610038161003816119891 (119905)1003816100381610038161003816 119899

minus120574)119901(119905)minus1

=1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901(119905))

le1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901minus) = 119899

minus1198871003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

(45)

It follows from condition (15) that

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119899

minus119887intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905)

le 119899minus119887

119862119887119899119887

intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

(46)

So by the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119862119887 int

Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905) le 119862119887 (47)

Consequently

intΩ

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 120588 (Ω) (119862119887

119901minus+ 119862119887

119901+) (48)

Step 2 Estimating the Second Term of (44) By the conditionintΩ

|119870119899(119909 119905)|119889120588(119905) le 1198620 in (14) with 1198620 ge 1 we know by theHolder inequality

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)

119889120588 (119905)

times (intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905))

119901(119909)minus1

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

times 119889120588 (119905) 119889120588 (119909)

= 119862119901+minus1

0intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

(49)

where

119869119899 (119891 119909 119905) =1003816100381610038161003816119870119899 (119909 119905)

10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

119905 isin Ω119899119909 119909 isin Ω

(50)

For 119905 isin Ω119899119909 we have a bound |119891(119905)| le 119899120574 and the

restriction |119909 minus 119905| le 119899minus14 From the log-Holder continuity

of the exponent function 119901 there exists a constant 119860119901 onlydependent on 119901 such that

119901 (119909) minus 119901 (119905) le119860119901

minus log 119899minus14=

4119860119901

log 119899 (51)

Abstract and Applied Analysis 7

When |119891(119905)| ge 1 we find

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119909)minus119901(119905)

le (119899120574)4119860119901 log 119899

le 1198994120574119860119901 log 119899 le 119901 (52)

where the constant number 119901 defined by

119901 = sup119899isinN

1198994120574119860119901 log 119899 (53)

is finite because

log 1198994120574119860119901 log 119899 =

4120574119860119901 log 119899

log 119899997888rarr 4120574119860119901 as 119899 997888rarr infin

(54)

When |119891(119905)| lt 1 we simply use |119891(119905)|119901(119909)minus119901(119905)

le

|119891(119905)|minus119901(119905) Applying these bounds we can estimate the core

part of Step 2 as

intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

(119901 +1003816100381610038161003816119891 (119905)

1003816100381610038161003816minus119901(119905)

)

times 119889120588 (119905) 119889120588 (119909)

= 119901 intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

+ intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) 119889120588 (119909)

le 1199011198620 intΩ

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

+ 1198620120588 (Ω) le 1198620119901 + 1198620120588 (Ω)

(55)

Here we have used the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 and(14) So the second term of (44) can be estimated as

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+0

(119901 + 120588 (Ω)) (56)

Step 3 Estimating the Third Term of (44) For 119905 isin Ω1015840

119899119909 we

have |119891(119905)| le 119899120574 and |119909 minus 119905| gt 119899

minus14 yielding 11989914

|119909 minus 119905| gt 1 Itfollows that

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816 le int

Ω1015840119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119899

120574(119899

14|119909 minus 119905|)

2119903119901(119909)

119889120588 (119905)

(57)

where 119903 is the integer part of 2120574119901+ + 1 Applying the Holderinequality and (14) we see that

10038161003816100381610038161198711198993(119891 119909)1003816100381610038161003816119901(119909)

le intΩ1015840119899119909

119899120574119901(119909)+(1199032) 1003816100381610038161003816119870119899 (119909 119905)

1003816100381610038161003816 |119909 minus 119905|2119903

119889120588 (119905) 119862119901(119909)minus1

0

(58)

So by the bound 119901(119909) le 119901+ and condition (16) the third termof (44) can be bounded as

intΩ

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+minus1

0119899120574119901++(1199032)119862119903119899

minus119903

= 119862119901+minus1

0119862119903119899

120574119901+minus(1199032) le 119862119901+minus1

0119862119903

(59)

Finally we put the estimates (48) (56) and (59) into (44)to conclude

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(60)

Take

120582 = 119872119901119887 = 1 + 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(61)

we find

intΩ

(

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le (1

120582)

119901minus

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 1

(62)

This implies 119871119899(119891)119871119901(sdot)

120588

le 119872119901119887 The bound 119872119901119887 isindependent of 119891 So we have 119871119899 le 119872119901119887 The proofTheorem 6 is complete

We are now in a position to proveTheorem 7

Proof of Theorem 7 We follow the standard procedure inapproximation theory and consider the error 119871119899(119892 119909) minus 119892(119909)

for 119892 isin 119882119903infin

119901 Apply the Taylor expansion

119892 (119905) = 119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119909 119905 isin Ω

(63)

where the remainder term 119877119892119903(119909 119905) is given by

119877119892119903 (119909 119905)

= int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572(119905 minus 119909)

120572119889119906

(64)

8 Abstract and Applied Analysis

We see from the vanishing moment condition (20) that

119871119899 (119892 119909) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119889120588 (119905) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572

times (119905 minus 119909)120572119889119906 119889120588 (119905)

(65)

SinceΩ is convex 119909+119906(119905minus119909) isin Ω for any 119906 isin [0 1] 119909 119905 isin ΩSo |119863

120572119892(119909 + 119906(119905 minus 119909))| le 119892

119901119903infinand we have

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

le int

1

0

(1 minus 119906)119903minus1

times sum

|120572|1=119903

10038171003817100381710038171198921003817100381710038171003817119901119903infin

120572int

Ω

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

|120572|119889120588 (119905) 119889119906

le 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588 (119905)

(66)

By (14) and the Holder inequality

(intΩ

1003816100381610038161003816119870119899(119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588(119905))

119901(119909)

le intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119901(119909)119889120588 (119905) 119862

119901(119909)minus1

0

(67)

If we denote the largest integer 119903minus satisfying 2119903minus le 119903119901minus as 119903minusand the smallest integer 119903+ satisfying 2119903+ ge 119903119901+ we find

|119905 minus 119909|119903119901(119909)

le |119905 minus 119909|

119903119901+ le |119905 minus 119909|2119903+ if |119905 minus 119909| ge 1

|119905 minus 119909|119903119901minus le |119905 minus 119909|

2119903minus if |119905 minus 119909| lt 1(68)

We combine this with (16) and see that for 120582 = 119889119903119892

119901119903infin

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903+ + |119905 minus 119909|2119903minus 119889120588 (119905) 119889120588 (119909)

le 119862119901+minus1

0(119862119903+

119899minus119903+ + 119862119903minus

119899minus119903minus) le 119862

119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus

(69)

When 119899 ge 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus we take

= 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin(119862119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus)1119901+ (70)

and find

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

)

119901(119909)

119889120588 (119909) le 1 (71)

which implies

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

le le 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

(72)

Thus by Theorem 6 and taking infimum over 119892 isin 119882119903infin

119901 we

have1003817100381710038171003817119871119899(119891) minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

1003817100381710038171003817119871119899(119891 minus 119892)

1003817100381710038171003817119871119901(sdot)120588+

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

+1003817100381710038171003817119892 minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

(119872119901119887 + 1)1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588

+1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot)

(73)

where the constant 119860119901119887119889 is given by

119860119901119887119889 = 119872119901119887 + 1 + 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(74)

When 119899 lt 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus then from the

inequality 119891 minus 119892119871119901(sdot)

120588

+ 119905119892119901119903infin

ge 119905119891119871119901(sdot)

120588

valid for 119905 le 1

we observe

K119903(119891 119899minus119903minus119901+)

119901(sdot)ge 119899

minus119903minus119901+10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

(75)

and applyingTheorem 6 directly yields

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1)1003817100381710038171003817119891

1003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1) 119899119903minus119901+K119903(119891 119899

minus119903minus119901+)119901(sdot)

(76)

and thereby (21) by setting

119860119901119887119889 = (119872119901119887 + 1) 1198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(77)

The proof of Theorem 7 is complete

Abstract and Applied Analysis 9

6 Further Topics and Discussion

Approximation by linear operators is an important topicin approximation theory It mainly consists of two familiesof approximation schemes Bernstein type positive linearoperators and quasi-interpolation type linear operators inmultivariate approximation

In this paper we mainly consider Bernstein type pos-itive linear operators We verified a conjecture in [11]about the uniformboundedness of Bernstein-Durrmeyer andBernstein-Kantorovich operators with respect to an arbitraryBorel measure on (0 1) on the variable 119871

119901(sdot)

120588space under

the assumption of log-Holder continuity of the exponentfunction 119901 We also provide quantitative estimates for highorders of approximation on the variable 119871

119901(sdot)

120588by linear

combinations of Bernstein type positive linear operatorsThe study of quasi-interpolation type linear operators

started with the classical work of Schoenberg on cardinalinterpolation by B-splines It has been developed significantlydue to important applications in the areas of finite elementmethods cardinal interpolation for multivariate approxima-tion and wavelet analysis A large class of linear operators forapproximating functions on R119889 take the form

119879 (119891 119909) = intR119889

Φ (119909 119905) 119891 (119905) 119889119905 119909 isin R119889 (78)

where Φ R119889times R119889

rarr R is a window functionsatisfying int

R119889Φ(119909 119905)119889119905 = 1 and some conditions for decays

of |Φ(119909 119905) as |119909 minus 119905| increases Quantitative estimates for theapproximation of functions in 119862(R119889

) cap 119871infin

(R119889) or 119871

119901(R119889

)

with 1 le 119901 lt infin can be found in a large literature ofmultivariate approximation (see eg [20 21 28]) Establish-ing analysis for approximation by quasi-interpolation typelinear operators on the variable 119871

119901(sdot)

120588spaces would be an

interesting topic An immediate barrier we meet with suchanalysis is the boundedness assumption of the measure 120588

(120588(Ω) lt infin) This assumption is not satisfied for mostquasi-interpolation type linear operators or the classicalWeierstrass (or Gaussian convolution) operators W119899(119891) =

(radic1198992120587)119889

intR119889

expminus119899119905 minus 1199092

22119891(119905)119889119905 for which 120588 is often

the Lebesgue measure on R119889 It is desirable to overcomethe technical difficulty and establish error analysis for linearoperators with respect to unbounded measures

We describedmotivations of our study in learning theoryIt would be interesting to implement detailed error analysisfor some related learning algorithms in classification andquantile regression by means of our results on orders ofapproximation by linear operators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their constructive suggestions and comments The work

described in this paper is supported partially by the ResearchGrants Council of Hong Kong (Project no CityU 105011)Thecorresponding author is Ding-Xuan Zhou

References

[1] S N Bernstein ldquoDemonstration du teoreme de Weirerstrassfondee sur le calcul des probabilitesrdquo Communications of theKharkov Mathematical Society vol 13 pp 1ndash2 1913

[2] L V Kantorovich ldquoSur certaines developments suivant lespolynomes de la forme de S Bernstein I-IIrdquo Comptes Rendus delrsquoAcademie des Sciences de LrsquoURSS A vol 563ndash568 pp 595ndash6001930

[3] J L Durrmeyer Une formule drsquoinversion de la transformeeLaplace applications a la theorie des moments [These de 3e cycleSciences] Faculte des Sciences lrsquoUniversite Paris Paris France1967

[4] H Berens and G G Lorentz ldquoInverse theorems for Bernsteinpolynomialsrdquo Indiana University Mathematics Journal vol 21pp 693ndash708 1972

[5] H Berens and R A DeVore ldquoQuantitative Korovkin theoremsfor positive linear operators on 119871119875-spacesrdquo Transactions of theAmerican Mathematical Society vol 245 pp 349ndash361 1978

[6] Z Ditzian and V TotikModuli of Smoothness vol 9 of SpringerSeries in Computational Mathematics Springer New York NYUSA 1987

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents Springer BerlinGermany 2011

[8] W Orlicz ldquoUber konjugierte Exponentenfolgenrdquo Studia Math-ematica vol 3 pp 200ndash211 1931

[9] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[10] O Kovacik and J Rakosnk ldquoOn spaces 119871119901(119909) and 119882

1119901(119909)rdquoCzechoslovakMathematical Journal vol 41 no 116 pp 592ndash6181991

[11] D X Zhou ldquoApproximation by positive linear operators onvariables 119871

119901(119909) spacesrdquo Journal of Applied Functional Analysisvol 9 no 3-4 pp 379ndash391 2014

[12] D X Zhou and K Jetter ldquoApproximation with polynomialkernels and SVM classifiersrdquo Advances in Computational Math-ematics vol 25 no 1ndash3 pp 323ndash344 2006

[13] E E Berdysheva and K Jetter ldquoMultivariate Bernstein-Durrmeyer operators with arbitrary weight functionsrdquo Journalof Approximation Theory vol 162 no 3 pp 576ndash598 2010

[14] A B Tsybakov ldquoOptimal aggregation of classifiers in statisticallearningrdquo The Annals of Statistics vol 32 no 1 pp 135ndash1662004

[15] S Smale andD X Zhou ldquoLearning theory estimates via integraloperators and their approximationsrdquo Constructive Approxima-tion vol 26 no 2 pp 153ndash172 2007

[16] S Smale and D X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquoThe American MathematicalSociety Bulletin vol 41 no 3 pp 279ndash305 2004

[17] T Hu J Fan Q Wu and D X Zhou ldquoRegularization schemesfor minimum error entropy principlerdquo Analysis and Applica-tions 2014

[18] I Steinwart and A Christmann ldquoEstimating conditional quan-tiles with the help of the pinball lossrdquo Bernoulli vol 17 no 1 pp211ndash225 2011

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Our second theorem gives orders of approximation whenthe approximated function has some smoothness stated interms of a K-functional Define a Holder space with index119903 isin N on Ω by

119882119903infin

119901= 119892 isin 119871

119901(sdot)

1205881003817100381710038171003817119892

1003817100381710038171003817119901119903infin lt infin (18)

where 119892119901119903infin

is the norm given by 119892119901119903infin

= 119892119871119901(sdot)

120588

+

sum|120572|1le119903119863

120572119892

infinwith 119863

120572119892 = 120597

|120572|112059711990912057211

sdot sdot sdot 120597119909120572119889

119889and |120572|1 =

sum119889

119895=1120572119895 for 120572 = (1205721 120572119889) isin Z119889

+ The K-functional

K119903(119891 119905)119901(sdot) is defined by

K119903(119891 119905)119901(sdot)

= inf119892isin119882119903infin119901

1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588+ 119905

10038171003817100381710038171198921003817100381710038171003817119901119903infin 119905 gt 0

(19)

Denote 119862infin

0(Ω) as the space of all compactly supported

119862infin functions on Ω From [7] we know that when 119901

+lt

infin 119862infin

0(Ω) is dense in 119871

119901(sdot)

120588(Ω) Hence for any 119891 isin 119871

119901(sdot)

120588(Ω)

there holdsK119903(119891 119905)119901(sdot) rarr 0 as 119905 rarr 0The following theorem to be proved in Section 5 and

extending the results for 119903 = 1 in [11] gives orders ofapproximation by linear operators on 119871

119901(sdot)

120588(Ω) when the K-

functional has explicit decay rates

Theorem 7 Under the assumption of Theorem 6 if Ω isconvex 119903 isin N and the kernels satisfy 119903119901minus ge 2 and

intΩ

119870119899 (119909 119905) (119905 minus 119909)120572119889120588 (119905) = 1205751205720 forall120572 isin Z

119889119908119894119905ℎ |120572|1 lt 119903

(20)

for almost every 119909 isin Ω then there holds for any 119891 isin 119871119901(sdot)

120588

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) (21)

where 119903minus is the integer part of 119903119901minus2 and the constant 119860119901119887119889 isindependent of 119891 isin 119871

119901(sdot)

120588(given explicitly in the proof)

The vanishing moment assumption (20) corresponds toStrang-Fix type conditions in the literature of shift-invariantspaces for example [20 21] It has appeared in the literatureof Bernstein type operators when linear combinations areconsidered as described by (34) in the next section

4 Approximation by Bernstein Type Operators

In this section we apply our main results to Bernstein typepositive linear operators and give high orders of approxima-tion by linear combinations of these operators on variable119871119901(sdot)

120588(Ω) spaces We demonstrate the analysis for the general

Bernstein-Durrmeyer operators in detail and describe brieflyresults for the general Bernstein-Kantorovich operators as anexample of other families of operators

The Bernstein-Durrmeyer operators on an open simplex

Ω = 119878 = 119909 isin R119889

+ |119909|1 lt 1 (22)

associated with a general positive Borel measure 120588 on 119878 aredefined as

119863119899 (119891 119909) = sum

|120572|1le119899

int119878

119891 (119905) 119901119899120572 (119905) 119889120588 (119905)

int119878

119901119899120572 (119905) 119889120588 (119905)119901119899120572 (119909)

119891 isin 1198711

120588 119909 isin 119878

(23)

where for 119909 = (1199091 119909119889) isin 119878 sub R119889 120572 = (1205721 120572119889) isin Z119889

+

we denote

119901119899120572 (119909) =119899

Π119889119895=1

120572119895 (119899 minus |120572|1)

119889

prod

119895=1

119909120572119895

119895(1 minus |119909|1)

119899minus|120572|1 (24)

The classical Bernstein-Durrmeyer operators (2) on Ω =

(0 1) (119889 = 1) with 119889120588(119909) = 119889119909 have been well studied (eg[22]) and extended to a multivariate form with respect toJacobi weights 119889120588(119909) = Π

119889

119895=1119909119902119895

119895119889119909 (eg [23]) Bernstein-

Durrmeyer operators on Ω = (0 1) with respect to anarbitrary Borel probability measure were introduced in [12]and applied to error analysis of learning algorithms forsupport vectormachine classificationsThemultidimensionalversion of such linear operators (23)was introduced in [13] In[24] the first author showed for a constant exponent function119901(119909) equiv 119901 isin [1 infin) that lim119899rarrinfin119891 minus 119863119899(119891)

119871119901

120588= 0 for any

119891 isin 119871119901

120588 The case 119901 equiv infin was studied in [25 26] Here we

consider the case with a general exponent function satisfying1 lt 119901minus lt 119901+ lt infin and the log-Holder continuity condition(4)

By applyingTheorem 6we can prove the uniformbound-edness of the Bernstein-Durrmeyer operators (23)

Proposition 8 LetΩ = 119878 120588 be a Borel probability measure on119878 and an exponent function 119901 119878 rarr (1 infin) satisfy 1 lt 119901minus lt

119901+ lt infin and the log-Holder continuity condition (4) If thereexist positive constants 119887 and 119862

lowast

119887such that int

119878119901119899120572(119905)119889120588(119905) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Durrmeyer operators defined on 119871119901(sdot)

120588(Ω) by (23) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119863119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (25)

Proof Define a sequence of kernels Ψ119899 on 119878 times 119878 by

Ψ119899 (119909 119905) = sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905) (26)

Then the Bernstein-Durrmeyer operators (23) can be writtenas

119863119899 (119891 119909) = int119878

Ψ119899 (119909 119905) 119891 (119905) 119889120588 (119905) (27)

So we only need to check the three conditions (14) (15) and(16) of Theorem 6

Abstract and Applied Analysis 5

Since Ψ119899(119909 119905) ge 0 we know that

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 119889120588 (119909) = int

119878

Ψ119899 (119909 119905) 119889120588 (119909)

= sum

|120572|1le119899

119901119899120572 (119905) int119878

119901119899120572 (119909) 119889120588 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

= sum

|120572|1le119899

119901119899120572 (119905) equiv 1

(28)

The same is true for int119878

|Ψ119899(119909 119905)|119889120588(119905) equiv 1 So we know thatcondition (14) holds with the constant 1198620 = 1

Applying the lower bound int119878

119901119899120572(119905)119889120588(119905) ge 119862lowast

119887119899minus119887 and

the inequality 119901119899120572(119905) le 1 we see that for any 119899 isin N and119909 119905 isin 119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 le sum

|120572|1le119899

119901119899120572 (119909)

119862lowast

119887119899minus119887

=1

119862lowast

119887

119899119887 (29)

Hence condition (15) holds with 119862119887 = 1119862lowast

119887

As for the last condition we separate 119905 minus 119909 into 119905 minus (120572119899) +

(120572119899) minus 119909 and find that for any 119903 isin N and 119899 isin N

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816|119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909)

le 22119903

int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

+ int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

= 22119903

int119878

sum

|120572|1le119899

119901119899120572 (119905)1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905)

+ int119878

sum

|120572|1le119899

119901119899120572 (119909)1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119909)

= 22119903+1

int119878

119861119899 ((sdot minus 119909)2119903

119909) 119889120588 (119909)

(30)

where 119861119899 is the multidimensional Bernstein operators on theclosure 119878 of 119878 defined by

119861119899 (119891 119909) = sum

|120572|1le119899

119891 (120572

119899) 119901119899120572 (119909) 119909 isin 119878 119891 isin 119862 (119878) (31)

It is well known [6] for the multidimensional Bernsteinoperators that there exists a constant 119862119889119903 depending only on119889 and 119903 such that

119861119899 ((sdot minus 119909)2119903

119909) le 119862119889119903119899minus119903

forall119909 isin 119878 (32)

It follows that

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909) le 2

2119903+1119862119889119903119899

minus119903 (33)

and condition (16) holds true with 119862119903 = 22119903+1

119862119889119903With all the three conditions verified the desired uniform

bound (25) for the Bernstein-Durrmeyer operators followsfromTheorem 6 This proves the proposition

The Bernstein-Durrmeyer operators (23) are positivewhich prevent from achieving high order approximation dueto a saturation phenomenon Linear combinations of suchoperators can be used to get high orders of approximationThe idea and literature review of this method can be found in[6] while further developments will not be mentioned hereThe linear combinations are defined as

119871119899119903 (119891 119909) =

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894(119891 119909) (34)

where 119898119889119903 = (119889 + 119903 minus 1)119889(119903 minus 1) is the dimension of thespace of polynomials of degree at most 119903 minus 1 and with twopositive constants 1198611 1198612 independent of 119899 we have

119899 = 1198990 lt 1198991 lt sdot sdot sdot lt 119899119898119889119903le 1198611119899

119898119889119903

sum

119894=0

1003816100381610038161003816119862119894 (119899)1003816100381610038161003816 le 1198612

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894((sdot minus 119909)

120572 119909) = 1205751205720 forall0 le |120572|1 le 119903 minus 1

(35)

For the classical Bernstein-Durrmeyer operators with respectto the Lebesgue measure (or even the Jacobi weights) theexistence of the above linear combinations can be seenand found in the literature The existence of such linearcombinations with respect to the arbitrary measure 120588 isa nontrivial problem and deserves intensive study Thistechnical question is out of the scope of this paper and willbe discussed in our further work Here we concentrate on thevariable 119871

119901(sdot)

120588(Ω) spaces and state the following result for the

high orders of approximation under the condition (35) whichis an immediate consequence of Theorem 7

Proposition 9 Under the assumption of Proposition 8 if 2 le

119903 isin 119873 and the operators 119871119899119903119899isinN defined by (34) satisfy (35)then for any 119891 isin 119871

119901(sdot)

120588 we have

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) (36)

where 119903minus is the integer part of 119903119901minus2 and the constant 119860119901119887119889 isindependent of 119891 isin 119871

119901(sdot)

120588

Let us now briefly describe approximation results for theBernstein-Kantorovich operators on 119878 defined [27] as

119861119870119899 (119891 119909) = sum

|120572|1le119899

int119878119899120572

119891 (119905) 119889120588 (119905)

120588 (119878119899120572)119901119899120572 (119909) 119891 isin 119871

1

120588 119909 isin 119878

(37)

6 Abstract and Applied Analysis

where 119878119899120572120572 are subdomains of 119878 defined by

119878119899120572 = 119909 isin 119878 119909 isin

119889

prod

119894=1

[120572119894

119899 + 1

120572119894

119899 + 1) |119909|1 le

|120572|1 + 1

119899 + 1

|120572|1 le 119899

(38)

In the same way as for the Bernstein-Durrmeyer oper-ators we have the following results for the Bernstein-Kantorovich operators

Proposition 10 Under the assumption of Proposition 8 for 119901if there exist positive constants 119887 and 119862

lowast

119887such that 120588(119878119899120572) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Kantorovich operators defined on 119871119901(sdot)

120588(Ω) by (37) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119861119870119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (39)

If 2 le 119903 isin 119873 and with 119863119899119894replaced by 119861119870119899119894

the operators119871119899119903119899isinN defined by (34) satisfy (35) then

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) forall119891 isin 119871

119901(sdot)

120588

(40)

5 Proof of Main Results

In this section we give detailed proof of our main results Letus first proveTheorem 6

Proof of Theorem 6 Let 119891 isin 119871119901(sdot) have norm 1 which implies

intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 Choose 120574 = 119887(119901minus minus 1) gt 0For 119909 isin Ω we define two subsets Ω119899119909 and Ω

1015840

119899119909of Ω as

Ω119899119909 = 119905 isin Ω 1003816100381610038161003816119891 (119905)

1003816100381610038161003816 le 119899120574 |119905 minus 119909| le 119899

minus14

Ω1015840

119899119909= 119905 isin Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le 119899

120574 |119905 minus 119909| gt 119899

minus14

(41)

Set

1198711198991 (119891 119909) = intΩ(Ω119899119909cupΩ

1015840119899119909)

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198992 (119891 119909) = intΩ119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198993 (119891 119909) = intΩ1015840119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

(42)

Then the value 119871119899(119891 119909) can be decomposed into three partsas

119871119899 (119891 119909) = 1198711198991 (119891 119909) + 1198711198992 (119891 119909) + 1198711198993 (119891 119909) 119909 isin Ω

(43)

and we have

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 3119901+

3

sum

119895=1

intΩ

10038161003816100381610038161003816119871119899119895 (119891 119909)

10038161003816100381610038161003816

119901(119909)

119889120588 (119909)

(44)

In the followingwe estimate the three terms in (44) separately

Step 1 Estimating the First Term of (44) By the definition of119901minus we have 119901(119905) ge 119901minus gt 1 For 119905 isin Ω (Ω119899119909 cup Ω

1015840

119899119909) we have

|119891(119905)| gt 119899120574 and thereby

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le

1003816100381610038161003816119891 (119905)1003816100381610038161003816 (

1003816100381610038161003816119891 (119905)1003816100381610038161003816 119899

minus120574)119901(119905)minus1

=1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901(119905))

le1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901minus) = 119899

minus1198871003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

(45)

It follows from condition (15) that

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119899

minus119887intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905)

le 119899minus119887

119862119887119899119887

intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

(46)

So by the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119862119887 int

Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905) le 119862119887 (47)

Consequently

intΩ

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 120588 (Ω) (119862119887

119901minus+ 119862119887

119901+) (48)

Step 2 Estimating the Second Term of (44) By the conditionintΩ

|119870119899(119909 119905)|119889120588(119905) le 1198620 in (14) with 1198620 ge 1 we know by theHolder inequality

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)

119889120588 (119905)

times (intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905))

119901(119909)minus1

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

times 119889120588 (119905) 119889120588 (119909)

= 119862119901+minus1

0intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

(49)

where

119869119899 (119891 119909 119905) =1003816100381610038161003816119870119899 (119909 119905)

10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

119905 isin Ω119899119909 119909 isin Ω

(50)

For 119905 isin Ω119899119909 we have a bound |119891(119905)| le 119899120574 and the

restriction |119909 minus 119905| le 119899minus14 From the log-Holder continuity

of the exponent function 119901 there exists a constant 119860119901 onlydependent on 119901 such that

119901 (119909) minus 119901 (119905) le119860119901

minus log 119899minus14=

4119860119901

log 119899 (51)

Abstract and Applied Analysis 7

When |119891(119905)| ge 1 we find

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119909)minus119901(119905)

le (119899120574)4119860119901 log 119899

le 1198994120574119860119901 log 119899 le 119901 (52)

where the constant number 119901 defined by

119901 = sup119899isinN

1198994120574119860119901 log 119899 (53)

is finite because

log 1198994120574119860119901 log 119899 =

4120574119860119901 log 119899

log 119899997888rarr 4120574119860119901 as 119899 997888rarr infin

(54)

When |119891(119905)| lt 1 we simply use |119891(119905)|119901(119909)minus119901(119905)

le

|119891(119905)|minus119901(119905) Applying these bounds we can estimate the core

part of Step 2 as

intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

(119901 +1003816100381610038161003816119891 (119905)

1003816100381610038161003816minus119901(119905)

)

times 119889120588 (119905) 119889120588 (119909)

= 119901 intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

+ intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) 119889120588 (119909)

le 1199011198620 intΩ

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

+ 1198620120588 (Ω) le 1198620119901 + 1198620120588 (Ω)

(55)

Here we have used the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 and(14) So the second term of (44) can be estimated as

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+0

(119901 + 120588 (Ω)) (56)

Step 3 Estimating the Third Term of (44) For 119905 isin Ω1015840

119899119909 we

have |119891(119905)| le 119899120574 and |119909 minus 119905| gt 119899

minus14 yielding 11989914

|119909 minus 119905| gt 1 Itfollows that

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816 le int

Ω1015840119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119899

120574(119899

14|119909 minus 119905|)

2119903119901(119909)

119889120588 (119905)

(57)

where 119903 is the integer part of 2120574119901+ + 1 Applying the Holderinequality and (14) we see that

10038161003816100381610038161198711198993(119891 119909)1003816100381610038161003816119901(119909)

le intΩ1015840119899119909

119899120574119901(119909)+(1199032) 1003816100381610038161003816119870119899 (119909 119905)

1003816100381610038161003816 |119909 minus 119905|2119903

119889120588 (119905) 119862119901(119909)minus1

0

(58)

So by the bound 119901(119909) le 119901+ and condition (16) the third termof (44) can be bounded as

intΩ

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+minus1

0119899120574119901++(1199032)119862119903119899

minus119903

= 119862119901+minus1

0119862119903119899

120574119901+minus(1199032) le 119862119901+minus1

0119862119903

(59)

Finally we put the estimates (48) (56) and (59) into (44)to conclude

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(60)

Take

120582 = 119872119901119887 = 1 + 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(61)

we find

intΩ

(

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le (1

120582)

119901minus

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 1

(62)

This implies 119871119899(119891)119871119901(sdot)

120588

le 119872119901119887 The bound 119872119901119887 isindependent of 119891 So we have 119871119899 le 119872119901119887 The proofTheorem 6 is complete

We are now in a position to proveTheorem 7

Proof of Theorem 7 We follow the standard procedure inapproximation theory and consider the error 119871119899(119892 119909) minus 119892(119909)

for 119892 isin 119882119903infin

119901 Apply the Taylor expansion

119892 (119905) = 119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119909 119905 isin Ω

(63)

where the remainder term 119877119892119903(119909 119905) is given by

119877119892119903 (119909 119905)

= int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572(119905 minus 119909)

120572119889119906

(64)

8 Abstract and Applied Analysis

We see from the vanishing moment condition (20) that

119871119899 (119892 119909) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119889120588 (119905) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572

times (119905 minus 119909)120572119889119906 119889120588 (119905)

(65)

SinceΩ is convex 119909+119906(119905minus119909) isin Ω for any 119906 isin [0 1] 119909 119905 isin ΩSo |119863

120572119892(119909 + 119906(119905 minus 119909))| le 119892

119901119903infinand we have

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

le int

1

0

(1 minus 119906)119903minus1

times sum

|120572|1=119903

10038171003817100381710038171198921003817100381710038171003817119901119903infin

120572int

Ω

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

|120572|119889120588 (119905) 119889119906

le 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588 (119905)

(66)

By (14) and the Holder inequality

(intΩ

1003816100381610038161003816119870119899(119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588(119905))

119901(119909)

le intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119901(119909)119889120588 (119905) 119862

119901(119909)minus1

0

(67)

If we denote the largest integer 119903minus satisfying 2119903minus le 119903119901minus as 119903minusand the smallest integer 119903+ satisfying 2119903+ ge 119903119901+ we find

|119905 minus 119909|119903119901(119909)

le |119905 minus 119909|

119903119901+ le |119905 minus 119909|2119903+ if |119905 minus 119909| ge 1

|119905 minus 119909|119903119901minus le |119905 minus 119909|

2119903minus if |119905 minus 119909| lt 1(68)

We combine this with (16) and see that for 120582 = 119889119903119892

119901119903infin

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903+ + |119905 minus 119909|2119903minus 119889120588 (119905) 119889120588 (119909)

le 119862119901+minus1

0(119862119903+

119899minus119903+ + 119862119903minus

119899minus119903minus) le 119862

119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus

(69)

When 119899 ge 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus we take

= 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin(119862119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus)1119901+ (70)

and find

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

)

119901(119909)

119889120588 (119909) le 1 (71)

which implies

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

le le 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

(72)

Thus by Theorem 6 and taking infimum over 119892 isin 119882119903infin

119901 we

have1003817100381710038171003817119871119899(119891) minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

1003817100381710038171003817119871119899(119891 minus 119892)

1003817100381710038171003817119871119901(sdot)120588+

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

+1003817100381710038171003817119892 minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

(119872119901119887 + 1)1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588

+1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot)

(73)

where the constant 119860119901119887119889 is given by

119860119901119887119889 = 119872119901119887 + 1 + 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(74)

When 119899 lt 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus then from the

inequality 119891 minus 119892119871119901(sdot)

120588

+ 119905119892119901119903infin

ge 119905119891119871119901(sdot)

120588

valid for 119905 le 1

we observe

K119903(119891 119899minus119903minus119901+)

119901(sdot)ge 119899

minus119903minus119901+10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

(75)

and applyingTheorem 6 directly yields

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1)1003817100381710038171003817119891

1003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1) 119899119903minus119901+K119903(119891 119899

minus119903minus119901+)119901(sdot)

(76)

and thereby (21) by setting

119860119901119887119889 = (119872119901119887 + 1) 1198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(77)

The proof of Theorem 7 is complete

Abstract and Applied Analysis 9

6 Further Topics and Discussion

Approximation by linear operators is an important topicin approximation theory It mainly consists of two familiesof approximation schemes Bernstein type positive linearoperators and quasi-interpolation type linear operators inmultivariate approximation

In this paper we mainly consider Bernstein type pos-itive linear operators We verified a conjecture in [11]about the uniformboundedness of Bernstein-Durrmeyer andBernstein-Kantorovich operators with respect to an arbitraryBorel measure on (0 1) on the variable 119871

119901(sdot)

120588space under

the assumption of log-Holder continuity of the exponentfunction 119901 We also provide quantitative estimates for highorders of approximation on the variable 119871

119901(sdot)

120588by linear

combinations of Bernstein type positive linear operatorsThe study of quasi-interpolation type linear operators

started with the classical work of Schoenberg on cardinalinterpolation by B-splines It has been developed significantlydue to important applications in the areas of finite elementmethods cardinal interpolation for multivariate approxima-tion and wavelet analysis A large class of linear operators forapproximating functions on R119889 take the form

119879 (119891 119909) = intR119889

Φ (119909 119905) 119891 (119905) 119889119905 119909 isin R119889 (78)

where Φ R119889times R119889

rarr R is a window functionsatisfying int

R119889Φ(119909 119905)119889119905 = 1 and some conditions for decays

of |Φ(119909 119905) as |119909 minus 119905| increases Quantitative estimates for theapproximation of functions in 119862(R119889

) cap 119871infin

(R119889) or 119871

119901(R119889

)

with 1 le 119901 lt infin can be found in a large literature ofmultivariate approximation (see eg [20 21 28]) Establish-ing analysis for approximation by quasi-interpolation typelinear operators on the variable 119871

119901(sdot)

120588spaces would be an

interesting topic An immediate barrier we meet with suchanalysis is the boundedness assumption of the measure 120588

(120588(Ω) lt infin) This assumption is not satisfied for mostquasi-interpolation type linear operators or the classicalWeierstrass (or Gaussian convolution) operators W119899(119891) =

(radic1198992120587)119889

intR119889

expminus119899119905 minus 1199092

22119891(119905)119889119905 for which 120588 is often

the Lebesgue measure on R119889 It is desirable to overcomethe technical difficulty and establish error analysis for linearoperators with respect to unbounded measures

We describedmotivations of our study in learning theoryIt would be interesting to implement detailed error analysisfor some related learning algorithms in classification andquantile regression by means of our results on orders ofapproximation by linear operators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their constructive suggestions and comments The work

described in this paper is supported partially by the ResearchGrants Council of Hong Kong (Project no CityU 105011)Thecorresponding author is Ding-Xuan Zhou

References

[1] S N Bernstein ldquoDemonstration du teoreme de Weirerstrassfondee sur le calcul des probabilitesrdquo Communications of theKharkov Mathematical Society vol 13 pp 1ndash2 1913

[2] L V Kantorovich ldquoSur certaines developments suivant lespolynomes de la forme de S Bernstein I-IIrdquo Comptes Rendus delrsquoAcademie des Sciences de LrsquoURSS A vol 563ndash568 pp 595ndash6001930

[3] J L Durrmeyer Une formule drsquoinversion de la transformeeLaplace applications a la theorie des moments [These de 3e cycleSciences] Faculte des Sciences lrsquoUniversite Paris Paris France1967

[4] H Berens and G G Lorentz ldquoInverse theorems for Bernsteinpolynomialsrdquo Indiana University Mathematics Journal vol 21pp 693ndash708 1972

[5] H Berens and R A DeVore ldquoQuantitative Korovkin theoremsfor positive linear operators on 119871119875-spacesrdquo Transactions of theAmerican Mathematical Society vol 245 pp 349ndash361 1978

[6] Z Ditzian and V TotikModuli of Smoothness vol 9 of SpringerSeries in Computational Mathematics Springer New York NYUSA 1987

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents Springer BerlinGermany 2011

[8] W Orlicz ldquoUber konjugierte Exponentenfolgenrdquo Studia Math-ematica vol 3 pp 200ndash211 1931

[9] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[10] O Kovacik and J Rakosnk ldquoOn spaces 119871119901(119909) and 119882

1119901(119909)rdquoCzechoslovakMathematical Journal vol 41 no 116 pp 592ndash6181991

[11] D X Zhou ldquoApproximation by positive linear operators onvariables 119871

119901(119909) spacesrdquo Journal of Applied Functional Analysisvol 9 no 3-4 pp 379ndash391 2014

[12] D X Zhou and K Jetter ldquoApproximation with polynomialkernels and SVM classifiersrdquo Advances in Computational Math-ematics vol 25 no 1ndash3 pp 323ndash344 2006

[13] E E Berdysheva and K Jetter ldquoMultivariate Bernstein-Durrmeyer operators with arbitrary weight functionsrdquo Journalof Approximation Theory vol 162 no 3 pp 576ndash598 2010

[14] A B Tsybakov ldquoOptimal aggregation of classifiers in statisticallearningrdquo The Annals of Statistics vol 32 no 1 pp 135ndash1662004

[15] S Smale andD X Zhou ldquoLearning theory estimates via integraloperators and their approximationsrdquo Constructive Approxima-tion vol 26 no 2 pp 153ndash172 2007

[16] S Smale and D X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquoThe American MathematicalSociety Bulletin vol 41 no 3 pp 279ndash305 2004

[17] T Hu J Fan Q Wu and D X Zhou ldquoRegularization schemesfor minimum error entropy principlerdquo Analysis and Applica-tions 2014

[18] I Steinwart and A Christmann ldquoEstimating conditional quan-tiles with the help of the pinball lossrdquo Bernoulli vol 17 no 1 pp211ndash225 2011

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Since Ψ119899(119909 119905) ge 0 we know that

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 119889120588 (119909) = int

119878

Ψ119899 (119909 119905) 119889120588 (119909)

= sum

|120572|1le119899

119901119899120572 (119905) int119878

119901119899120572 (119909) 119889120588 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

= sum

|120572|1le119899

119901119899120572 (119905) equiv 1

(28)

The same is true for int119878

|Ψ119899(119909 119905)|119889120588(119905) equiv 1 So we know thatcondition (14) holds with the constant 1198620 = 1

Applying the lower bound int119878

119901119899120572(119905)119889120588(119905) ge 119862lowast

119887119899minus119887 and

the inequality 119901119899120572(119905) le 1 we see that for any 119899 isin N and119909 119905 isin 119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 le sum

|120572|1le119899

119901119899120572 (119909)

119862lowast

119887119899minus119887

=1

119862lowast

119887

119899119887 (29)

Hence condition (15) holds with 119862119887 = 1119862lowast

119887

As for the last condition we separate 119905 minus 119909 into 119905 minus (120572119899) +

(120572119899) minus 119909 and find that for any 119903 isin N and 119899 isin N

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816|119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909)

le 22119903

int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

+ int119878

int119878

sum

|120572|1le119899

119901119899120572 (119905) 119901119899120572 (119909)

int119878

119901119899120572 (119905) 119889120588 (119905)

times1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905) 119889120588 (119909)

= 22119903

int119878

sum

|120572|1le119899

119901119899120572 (119905)1003816100381610038161003816100381610038161003816119905 minus

120572

119899

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119905)

+ int119878

sum

|120572|1le119899

119901119899120572 (119909)1003816100381610038161003816100381610038161003816

120572

119899minus 119909

1003816100381610038161003816100381610038161003816

2119903

119889120588 (119909)

= 22119903+1

int119878

119861119899 ((sdot minus 119909)2119903

119909) 119889120588 (119909)

(30)

where 119861119899 is the multidimensional Bernstein operators on theclosure 119878 of 119878 defined by

119861119899 (119891 119909) = sum

|120572|1le119899

119891 (120572

119899) 119901119899120572 (119909) 119909 isin 119878 119891 isin 119862 (119878) (31)

It is well known [6] for the multidimensional Bernsteinoperators that there exists a constant 119862119889119903 depending only on119889 and 119903 such that

119861119899 ((sdot minus 119909)2119903

119909) le 119862119889119903119899minus119903

forall119909 isin 119878 (32)

It follows that

int119878

int119878

1003816100381610038161003816Ψ119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903119889120588 (119905) 119889120588 (119909) le 2

2119903+1119862119889119903119899

minus119903 (33)

and condition (16) holds true with 119862119903 = 22119903+1

119862119889119903With all the three conditions verified the desired uniform

bound (25) for the Bernstein-Durrmeyer operators followsfromTheorem 6 This proves the proposition

The Bernstein-Durrmeyer operators (23) are positivewhich prevent from achieving high order approximation dueto a saturation phenomenon Linear combinations of suchoperators can be used to get high orders of approximationThe idea and literature review of this method can be found in[6] while further developments will not be mentioned hereThe linear combinations are defined as

119871119899119903 (119891 119909) =

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894(119891 119909) (34)

where 119898119889119903 = (119889 + 119903 minus 1)119889(119903 minus 1) is the dimension of thespace of polynomials of degree at most 119903 minus 1 and with twopositive constants 1198611 1198612 independent of 119899 we have

119899 = 1198990 lt 1198991 lt sdot sdot sdot lt 119899119898119889119903le 1198611119899

119898119889119903

sum

119894=0

1003816100381610038161003816119862119894 (119899)1003816100381610038161003816 le 1198612

119898119889119903

sum

119894=0

119862119894 (119899) 119863119899119894((sdot minus 119909)

120572 119909) = 1205751205720 forall0 le |120572|1 le 119903 minus 1

(35)

For the classical Bernstein-Durrmeyer operators with respectto the Lebesgue measure (or even the Jacobi weights) theexistence of the above linear combinations can be seenand found in the literature The existence of such linearcombinations with respect to the arbitrary measure 120588 isa nontrivial problem and deserves intensive study Thistechnical question is out of the scope of this paper and willbe discussed in our further work Here we concentrate on thevariable 119871

119901(sdot)

120588(Ω) spaces and state the following result for the

high orders of approximation under the condition (35) whichis an immediate consequence of Theorem 7

Proposition 9 Under the assumption of Proposition 8 if 2 le

119903 isin 119873 and the operators 119871119899119903119899isinN defined by (34) satisfy (35)then for any 119891 isin 119871

119901(sdot)

120588 we have

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) (36)

where 119903minus is the integer part of 119903119901minus2 and the constant 119860119901119887119889 isindependent of 119891 isin 119871

119901(sdot)

120588

Let us now briefly describe approximation results for theBernstein-Kantorovich operators on 119878 defined [27] as

119861119870119899 (119891 119909) = sum

|120572|1le119899

int119878119899120572

119891 (119905) 119889120588 (119905)

120588 (119878119899120572)119901119899120572 (119909) 119891 isin 119871

1

120588 119909 isin 119878

(37)

6 Abstract and Applied Analysis

where 119878119899120572120572 are subdomains of 119878 defined by

119878119899120572 = 119909 isin 119878 119909 isin

119889

prod

119894=1

[120572119894

119899 + 1

120572119894

119899 + 1) |119909|1 le

|120572|1 + 1

119899 + 1

|120572|1 le 119899

(38)

In the same way as for the Bernstein-Durrmeyer oper-ators we have the following results for the Bernstein-Kantorovich operators

Proposition 10 Under the assumption of Proposition 8 for 119901if there exist positive constants 119887 and 119862

lowast

119887such that 120588(119878119899120572) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Kantorovich operators defined on 119871119901(sdot)

120588(Ω) by (37) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119861119870119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (39)

If 2 le 119903 isin 119873 and with 119863119899119894replaced by 119861119870119899119894

the operators119871119899119903119899isinN defined by (34) satisfy (35) then

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) forall119891 isin 119871

119901(sdot)

120588

(40)

5 Proof of Main Results

In this section we give detailed proof of our main results Letus first proveTheorem 6

Proof of Theorem 6 Let 119891 isin 119871119901(sdot) have norm 1 which implies

intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 Choose 120574 = 119887(119901minus minus 1) gt 0For 119909 isin Ω we define two subsets Ω119899119909 and Ω

1015840

119899119909of Ω as

Ω119899119909 = 119905 isin Ω 1003816100381610038161003816119891 (119905)

1003816100381610038161003816 le 119899120574 |119905 minus 119909| le 119899

minus14

Ω1015840

119899119909= 119905 isin Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le 119899

120574 |119905 minus 119909| gt 119899

minus14

(41)

Set

1198711198991 (119891 119909) = intΩ(Ω119899119909cupΩ

1015840119899119909)

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198992 (119891 119909) = intΩ119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198993 (119891 119909) = intΩ1015840119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

(42)

Then the value 119871119899(119891 119909) can be decomposed into three partsas

119871119899 (119891 119909) = 1198711198991 (119891 119909) + 1198711198992 (119891 119909) + 1198711198993 (119891 119909) 119909 isin Ω

(43)

and we have

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 3119901+

3

sum

119895=1

intΩ

10038161003816100381610038161003816119871119899119895 (119891 119909)

10038161003816100381610038161003816

119901(119909)

119889120588 (119909)

(44)

In the followingwe estimate the three terms in (44) separately

Step 1 Estimating the First Term of (44) By the definition of119901minus we have 119901(119905) ge 119901minus gt 1 For 119905 isin Ω (Ω119899119909 cup Ω

1015840

119899119909) we have

|119891(119905)| gt 119899120574 and thereby

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le

1003816100381610038161003816119891 (119905)1003816100381610038161003816 (

1003816100381610038161003816119891 (119905)1003816100381610038161003816 119899

minus120574)119901(119905)minus1

=1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901(119905))

le1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901minus) = 119899

minus1198871003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

(45)

It follows from condition (15) that

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119899

minus119887intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905)

le 119899minus119887

119862119887119899119887

intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

(46)

So by the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119862119887 int

Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905) le 119862119887 (47)

Consequently

intΩ

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 120588 (Ω) (119862119887

119901minus+ 119862119887

119901+) (48)

Step 2 Estimating the Second Term of (44) By the conditionintΩ

|119870119899(119909 119905)|119889120588(119905) le 1198620 in (14) with 1198620 ge 1 we know by theHolder inequality

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)

119889120588 (119905)

times (intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905))

119901(119909)minus1

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

times 119889120588 (119905) 119889120588 (119909)

= 119862119901+minus1

0intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

(49)

where

119869119899 (119891 119909 119905) =1003816100381610038161003816119870119899 (119909 119905)

10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

119905 isin Ω119899119909 119909 isin Ω

(50)

For 119905 isin Ω119899119909 we have a bound |119891(119905)| le 119899120574 and the

restriction |119909 minus 119905| le 119899minus14 From the log-Holder continuity

of the exponent function 119901 there exists a constant 119860119901 onlydependent on 119901 such that

119901 (119909) minus 119901 (119905) le119860119901

minus log 119899minus14=

4119860119901

log 119899 (51)

Abstract and Applied Analysis 7

When |119891(119905)| ge 1 we find

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119909)minus119901(119905)

le (119899120574)4119860119901 log 119899

le 1198994120574119860119901 log 119899 le 119901 (52)

where the constant number 119901 defined by

119901 = sup119899isinN

1198994120574119860119901 log 119899 (53)

is finite because

log 1198994120574119860119901 log 119899 =

4120574119860119901 log 119899

log 119899997888rarr 4120574119860119901 as 119899 997888rarr infin

(54)

When |119891(119905)| lt 1 we simply use |119891(119905)|119901(119909)minus119901(119905)

le

|119891(119905)|minus119901(119905) Applying these bounds we can estimate the core

part of Step 2 as

intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

(119901 +1003816100381610038161003816119891 (119905)

1003816100381610038161003816minus119901(119905)

)

times 119889120588 (119905) 119889120588 (119909)

= 119901 intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

+ intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) 119889120588 (119909)

le 1199011198620 intΩ

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

+ 1198620120588 (Ω) le 1198620119901 + 1198620120588 (Ω)

(55)

Here we have used the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 and(14) So the second term of (44) can be estimated as

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+0

(119901 + 120588 (Ω)) (56)

Step 3 Estimating the Third Term of (44) For 119905 isin Ω1015840

119899119909 we

have |119891(119905)| le 119899120574 and |119909 minus 119905| gt 119899

minus14 yielding 11989914

|119909 minus 119905| gt 1 Itfollows that

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816 le int

Ω1015840119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119899

120574(119899

14|119909 minus 119905|)

2119903119901(119909)

119889120588 (119905)

(57)

where 119903 is the integer part of 2120574119901+ + 1 Applying the Holderinequality and (14) we see that

10038161003816100381610038161198711198993(119891 119909)1003816100381610038161003816119901(119909)

le intΩ1015840119899119909

119899120574119901(119909)+(1199032) 1003816100381610038161003816119870119899 (119909 119905)

1003816100381610038161003816 |119909 minus 119905|2119903

119889120588 (119905) 119862119901(119909)minus1

0

(58)

So by the bound 119901(119909) le 119901+ and condition (16) the third termof (44) can be bounded as

intΩ

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+minus1

0119899120574119901++(1199032)119862119903119899

minus119903

= 119862119901+minus1

0119862119903119899

120574119901+minus(1199032) le 119862119901+minus1

0119862119903

(59)

Finally we put the estimates (48) (56) and (59) into (44)to conclude

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(60)

Take

120582 = 119872119901119887 = 1 + 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(61)

we find

intΩ

(

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le (1

120582)

119901minus

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 1

(62)

This implies 119871119899(119891)119871119901(sdot)

120588

le 119872119901119887 The bound 119872119901119887 isindependent of 119891 So we have 119871119899 le 119872119901119887 The proofTheorem 6 is complete

We are now in a position to proveTheorem 7

Proof of Theorem 7 We follow the standard procedure inapproximation theory and consider the error 119871119899(119892 119909) minus 119892(119909)

for 119892 isin 119882119903infin

119901 Apply the Taylor expansion

119892 (119905) = 119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119909 119905 isin Ω

(63)

where the remainder term 119877119892119903(119909 119905) is given by

119877119892119903 (119909 119905)

= int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572(119905 minus 119909)

120572119889119906

(64)

8 Abstract and Applied Analysis

We see from the vanishing moment condition (20) that

119871119899 (119892 119909) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119889120588 (119905) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572

times (119905 minus 119909)120572119889119906 119889120588 (119905)

(65)

SinceΩ is convex 119909+119906(119905minus119909) isin Ω for any 119906 isin [0 1] 119909 119905 isin ΩSo |119863

120572119892(119909 + 119906(119905 minus 119909))| le 119892

119901119903infinand we have

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

le int

1

0

(1 minus 119906)119903minus1

times sum

|120572|1=119903

10038171003817100381710038171198921003817100381710038171003817119901119903infin

120572int

Ω

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

|120572|119889120588 (119905) 119889119906

le 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588 (119905)

(66)

By (14) and the Holder inequality

(intΩ

1003816100381610038161003816119870119899(119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588(119905))

119901(119909)

le intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119901(119909)119889120588 (119905) 119862

119901(119909)minus1

0

(67)

If we denote the largest integer 119903minus satisfying 2119903minus le 119903119901minus as 119903minusand the smallest integer 119903+ satisfying 2119903+ ge 119903119901+ we find

|119905 minus 119909|119903119901(119909)

le |119905 minus 119909|

119903119901+ le |119905 minus 119909|2119903+ if |119905 minus 119909| ge 1

|119905 minus 119909|119903119901minus le |119905 minus 119909|

2119903minus if |119905 minus 119909| lt 1(68)

We combine this with (16) and see that for 120582 = 119889119903119892

119901119903infin

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903+ + |119905 minus 119909|2119903minus 119889120588 (119905) 119889120588 (119909)

le 119862119901+minus1

0(119862119903+

119899minus119903+ + 119862119903minus

119899minus119903minus) le 119862

119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus

(69)

When 119899 ge 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus we take

= 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin(119862119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus)1119901+ (70)

and find

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

)

119901(119909)

119889120588 (119909) le 1 (71)

which implies

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

le le 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

(72)

Thus by Theorem 6 and taking infimum over 119892 isin 119882119903infin

119901 we

have1003817100381710038171003817119871119899(119891) minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

1003817100381710038171003817119871119899(119891 minus 119892)

1003817100381710038171003817119871119901(sdot)120588+

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

+1003817100381710038171003817119892 minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

(119872119901119887 + 1)1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588

+1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot)

(73)

where the constant 119860119901119887119889 is given by

119860119901119887119889 = 119872119901119887 + 1 + 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(74)

When 119899 lt 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus then from the

inequality 119891 minus 119892119871119901(sdot)

120588

+ 119905119892119901119903infin

ge 119905119891119871119901(sdot)

120588

valid for 119905 le 1

we observe

K119903(119891 119899minus119903minus119901+)

119901(sdot)ge 119899

minus119903minus119901+10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

(75)

and applyingTheorem 6 directly yields

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1)1003817100381710038171003817119891

1003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1) 119899119903minus119901+K119903(119891 119899

minus119903minus119901+)119901(sdot)

(76)

and thereby (21) by setting

119860119901119887119889 = (119872119901119887 + 1) 1198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(77)

The proof of Theorem 7 is complete

Abstract and Applied Analysis 9

6 Further Topics and Discussion

Approximation by linear operators is an important topicin approximation theory It mainly consists of two familiesof approximation schemes Bernstein type positive linearoperators and quasi-interpolation type linear operators inmultivariate approximation

In this paper we mainly consider Bernstein type pos-itive linear operators We verified a conjecture in [11]about the uniformboundedness of Bernstein-Durrmeyer andBernstein-Kantorovich operators with respect to an arbitraryBorel measure on (0 1) on the variable 119871

119901(sdot)

120588space under

the assumption of log-Holder continuity of the exponentfunction 119901 We also provide quantitative estimates for highorders of approximation on the variable 119871

119901(sdot)

120588by linear

combinations of Bernstein type positive linear operatorsThe study of quasi-interpolation type linear operators

started with the classical work of Schoenberg on cardinalinterpolation by B-splines It has been developed significantlydue to important applications in the areas of finite elementmethods cardinal interpolation for multivariate approxima-tion and wavelet analysis A large class of linear operators forapproximating functions on R119889 take the form

119879 (119891 119909) = intR119889

Φ (119909 119905) 119891 (119905) 119889119905 119909 isin R119889 (78)

where Φ R119889times R119889

rarr R is a window functionsatisfying int

R119889Φ(119909 119905)119889119905 = 1 and some conditions for decays

of |Φ(119909 119905) as |119909 minus 119905| increases Quantitative estimates for theapproximation of functions in 119862(R119889

) cap 119871infin

(R119889) or 119871

119901(R119889

)

with 1 le 119901 lt infin can be found in a large literature ofmultivariate approximation (see eg [20 21 28]) Establish-ing analysis for approximation by quasi-interpolation typelinear operators on the variable 119871

119901(sdot)

120588spaces would be an

interesting topic An immediate barrier we meet with suchanalysis is the boundedness assumption of the measure 120588

(120588(Ω) lt infin) This assumption is not satisfied for mostquasi-interpolation type linear operators or the classicalWeierstrass (or Gaussian convolution) operators W119899(119891) =

(radic1198992120587)119889

intR119889

expminus119899119905 minus 1199092

22119891(119905)119889119905 for which 120588 is often

the Lebesgue measure on R119889 It is desirable to overcomethe technical difficulty and establish error analysis for linearoperators with respect to unbounded measures

We describedmotivations of our study in learning theoryIt would be interesting to implement detailed error analysisfor some related learning algorithms in classification andquantile regression by means of our results on orders ofapproximation by linear operators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their constructive suggestions and comments The work

described in this paper is supported partially by the ResearchGrants Council of Hong Kong (Project no CityU 105011)Thecorresponding author is Ding-Xuan Zhou

References

[1] S N Bernstein ldquoDemonstration du teoreme de Weirerstrassfondee sur le calcul des probabilitesrdquo Communications of theKharkov Mathematical Society vol 13 pp 1ndash2 1913

[2] L V Kantorovich ldquoSur certaines developments suivant lespolynomes de la forme de S Bernstein I-IIrdquo Comptes Rendus delrsquoAcademie des Sciences de LrsquoURSS A vol 563ndash568 pp 595ndash6001930

[3] J L Durrmeyer Une formule drsquoinversion de la transformeeLaplace applications a la theorie des moments [These de 3e cycleSciences] Faculte des Sciences lrsquoUniversite Paris Paris France1967

[4] H Berens and G G Lorentz ldquoInverse theorems for Bernsteinpolynomialsrdquo Indiana University Mathematics Journal vol 21pp 693ndash708 1972

[5] H Berens and R A DeVore ldquoQuantitative Korovkin theoremsfor positive linear operators on 119871119875-spacesrdquo Transactions of theAmerican Mathematical Society vol 245 pp 349ndash361 1978

[6] Z Ditzian and V TotikModuli of Smoothness vol 9 of SpringerSeries in Computational Mathematics Springer New York NYUSA 1987

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents Springer BerlinGermany 2011

[8] W Orlicz ldquoUber konjugierte Exponentenfolgenrdquo Studia Math-ematica vol 3 pp 200ndash211 1931

[9] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[10] O Kovacik and J Rakosnk ldquoOn spaces 119871119901(119909) and 119882

1119901(119909)rdquoCzechoslovakMathematical Journal vol 41 no 116 pp 592ndash6181991

[11] D X Zhou ldquoApproximation by positive linear operators onvariables 119871

119901(119909) spacesrdquo Journal of Applied Functional Analysisvol 9 no 3-4 pp 379ndash391 2014

[12] D X Zhou and K Jetter ldquoApproximation with polynomialkernels and SVM classifiersrdquo Advances in Computational Math-ematics vol 25 no 1ndash3 pp 323ndash344 2006

[13] E E Berdysheva and K Jetter ldquoMultivariate Bernstein-Durrmeyer operators with arbitrary weight functionsrdquo Journalof Approximation Theory vol 162 no 3 pp 576ndash598 2010

[14] A B Tsybakov ldquoOptimal aggregation of classifiers in statisticallearningrdquo The Annals of Statistics vol 32 no 1 pp 135ndash1662004

[15] S Smale andD X Zhou ldquoLearning theory estimates via integraloperators and their approximationsrdquo Constructive Approxima-tion vol 26 no 2 pp 153ndash172 2007

[16] S Smale and D X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquoThe American MathematicalSociety Bulletin vol 41 no 3 pp 279ndash305 2004

[17] T Hu J Fan Q Wu and D X Zhou ldquoRegularization schemesfor minimum error entropy principlerdquo Analysis and Applica-tions 2014

[18] I Steinwart and A Christmann ldquoEstimating conditional quan-tiles with the help of the pinball lossrdquo Bernoulli vol 17 no 1 pp211ndash225 2011

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

where 119878119899120572120572 are subdomains of 119878 defined by

119878119899120572 = 119909 isin 119878 119909 isin

119889

prod

119894=1

[120572119894

119899 + 1

120572119894

119899 + 1) |119909|1 le

|120572|1 + 1

119899 + 1

|120572|1 le 119899

(38)

In the same way as for the Bernstein-Durrmeyer oper-ators we have the following results for the Bernstein-Kantorovich operators

Proposition 10 Under the assumption of Proposition 8 for 119901if there exist positive constants 119887 and 119862

lowast

119887such that 120588(119878119899120572) ge

119862lowast

119887119899minus119887 for 119899 isin N and |120572|1 le 119899 then for the Bernstein-

Kantorovich operators defined on 119871119901(sdot)

120588(Ω) by (37) there exists

a positive constant 119872119901119887 depending only on 119901 119887 119862lowast

119887such that

1003817100381710038171003817119861119870119899

1003817100381710038171003817 le 119872119901119887 forall119899 isin N (39)

If 2 le 119903 isin 119873 and with 119863119899119894replaced by 119861119870119899119894

the operators119871119899119903119899isinN defined by (34) satisfy (35) then

1003817100381710038171003817119871119899119903(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot) forall119891 isin 119871

119901(sdot)

120588

(40)

5 Proof of Main Results

In this section we give detailed proof of our main results Letus first proveTheorem 6

Proof of Theorem 6 Let 119891 isin 119871119901(sdot) have norm 1 which implies

intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 Choose 120574 = 119887(119901minus minus 1) gt 0For 119909 isin Ω we define two subsets Ω119899119909 and Ω

1015840

119899119909of Ω as

Ω119899119909 = 119905 isin Ω 1003816100381610038161003816119891 (119905)

1003816100381610038161003816 le 119899120574 |119905 minus 119909| le 119899

minus14

Ω1015840

119899119909= 119905 isin Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le 119899

120574 |119905 minus 119909| gt 119899

minus14

(41)

Set

1198711198991 (119891 119909) = intΩ(Ω119899119909cupΩ

1015840119899119909)

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198992 (119891 119909) = intΩ119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

1198711198993 (119891 119909) = intΩ1015840119899119909

119870119899 (119909 119905) 119891 (119905) 119889120588 (119905)

(42)

Then the value 119871119899(119891 119909) can be decomposed into three partsas

119871119899 (119891 119909) = 1198711198991 (119891 119909) + 1198711198992 (119891 119909) + 1198711198993 (119891 119909) 119909 isin Ω

(43)

and we have

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 3119901+

3

sum

119895=1

intΩ

10038161003816100381610038161003816119871119899119895 (119891 119909)

10038161003816100381610038161003816

119901(119909)

119889120588 (119909)

(44)

In the followingwe estimate the three terms in (44) separately

Step 1 Estimating the First Term of (44) By the definition of119901minus we have 119901(119905) ge 119901minus gt 1 For 119905 isin Ω (Ω119899119909 cup Ω

1015840

119899119909) we have

|119891(119905)| gt 119899120574 and thereby

1003816100381610038161003816119891 (119905)1003816100381610038161003816 le

1003816100381610038161003816119891 (119905)1003816100381610038161003816 (

1003816100381610038161003816119891 (119905)1003816100381610038161003816 119899

minus120574)119901(119905)minus1

=1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901(119905))

le1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119899120574(1minus119901minus) = 119899

minus1198871003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

(45)

It follows from condition (15) that

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119899

minus119887intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905)

le 119899minus119887

119862119887119899119887

intΩ(Ω119899119909cupΩ

1015840119899119909)

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

(46)

So by the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816 le 119862119887 int

Ω

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905) le 119862119887 (47)

Consequently

intΩ

10038161003816100381610038161198711198991 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 120588 (Ω) (119862119887

119901minus+ 119862119887

119901+) (48)

Step 2 Estimating the Second Term of (44) By the conditionintΩ

|119870119899(119909 119905)|119889120588(119905) le 1198620 in (14) with 1198620 ge 1 we know by theHolder inequality

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)

119889120588 (119905)

times (intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905))

119901(119909)minus1

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

times 119889120588 (119905) 119889120588 (119909)

= 119862119901+minus1

0intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

(49)

where

119869119899 (119891 119909 119905) =1003816100381610038161003816119870119899 (119909 119905)

10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119909)minus119901(119905)

119905 isin Ω119899119909 119909 isin Ω

(50)

For 119905 isin Ω119899119909 we have a bound |119891(119905)| le 119899120574 and the

restriction |119909 minus 119905| le 119899minus14 From the log-Holder continuity

of the exponent function 119901 there exists a constant 119860119901 onlydependent on 119901 such that

119901 (119909) minus 119901 (119905) le119860119901

minus log 119899minus14=

4119860119901

log 119899 (51)

Abstract and Applied Analysis 7

When |119891(119905)| ge 1 we find

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119909)minus119901(119905)

le (119899120574)4119860119901 log 119899

le 1198994120574119860119901 log 119899 le 119901 (52)

where the constant number 119901 defined by

119901 = sup119899isinN

1198994120574119860119901 log 119899 (53)

is finite because

log 1198994120574119860119901 log 119899 =

4120574119860119901 log 119899

log 119899997888rarr 4120574119860119901 as 119899 997888rarr infin

(54)

When |119891(119905)| lt 1 we simply use |119891(119905)|119901(119909)minus119901(119905)

le

|119891(119905)|minus119901(119905) Applying these bounds we can estimate the core

part of Step 2 as

intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

(119901 +1003816100381610038161003816119891 (119905)

1003816100381610038161003816minus119901(119905)

)

times 119889120588 (119905) 119889120588 (119909)

= 119901 intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

+ intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) 119889120588 (119909)

le 1199011198620 intΩ

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

+ 1198620120588 (Ω) le 1198620119901 + 1198620120588 (Ω)

(55)

Here we have used the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 and(14) So the second term of (44) can be estimated as

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+0

(119901 + 120588 (Ω)) (56)

Step 3 Estimating the Third Term of (44) For 119905 isin Ω1015840

119899119909 we

have |119891(119905)| le 119899120574 and |119909 minus 119905| gt 119899

minus14 yielding 11989914

|119909 minus 119905| gt 1 Itfollows that

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816 le int

Ω1015840119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119899

120574(119899

14|119909 minus 119905|)

2119903119901(119909)

119889120588 (119905)

(57)

where 119903 is the integer part of 2120574119901+ + 1 Applying the Holderinequality and (14) we see that

10038161003816100381610038161198711198993(119891 119909)1003816100381610038161003816119901(119909)

le intΩ1015840119899119909

119899120574119901(119909)+(1199032) 1003816100381610038161003816119870119899 (119909 119905)

1003816100381610038161003816 |119909 minus 119905|2119903

119889120588 (119905) 119862119901(119909)minus1

0

(58)

So by the bound 119901(119909) le 119901+ and condition (16) the third termof (44) can be bounded as

intΩ

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+minus1

0119899120574119901++(1199032)119862119903119899

minus119903

= 119862119901+minus1

0119862119903119899

120574119901+minus(1199032) le 119862119901+minus1

0119862119903

(59)

Finally we put the estimates (48) (56) and (59) into (44)to conclude

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(60)

Take

120582 = 119872119901119887 = 1 + 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(61)

we find

intΩ

(

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le (1

120582)

119901minus

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 1

(62)

This implies 119871119899(119891)119871119901(sdot)

120588

le 119872119901119887 The bound 119872119901119887 isindependent of 119891 So we have 119871119899 le 119872119901119887 The proofTheorem 6 is complete

We are now in a position to proveTheorem 7

Proof of Theorem 7 We follow the standard procedure inapproximation theory and consider the error 119871119899(119892 119909) minus 119892(119909)

for 119892 isin 119882119903infin

119901 Apply the Taylor expansion

119892 (119905) = 119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119909 119905 isin Ω

(63)

where the remainder term 119877119892119903(119909 119905) is given by

119877119892119903 (119909 119905)

= int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572(119905 minus 119909)

120572119889119906

(64)

8 Abstract and Applied Analysis

We see from the vanishing moment condition (20) that

119871119899 (119892 119909) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119889120588 (119905) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572

times (119905 minus 119909)120572119889119906 119889120588 (119905)

(65)

SinceΩ is convex 119909+119906(119905minus119909) isin Ω for any 119906 isin [0 1] 119909 119905 isin ΩSo |119863

120572119892(119909 + 119906(119905 minus 119909))| le 119892

119901119903infinand we have

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

le int

1

0

(1 minus 119906)119903minus1

times sum

|120572|1=119903

10038171003817100381710038171198921003817100381710038171003817119901119903infin

120572int

Ω

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

|120572|119889120588 (119905) 119889119906

le 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588 (119905)

(66)

By (14) and the Holder inequality

(intΩ

1003816100381610038161003816119870119899(119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588(119905))

119901(119909)

le intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119901(119909)119889120588 (119905) 119862

119901(119909)minus1

0

(67)

If we denote the largest integer 119903minus satisfying 2119903minus le 119903119901minus as 119903minusand the smallest integer 119903+ satisfying 2119903+ ge 119903119901+ we find

|119905 minus 119909|119903119901(119909)

le |119905 minus 119909|

119903119901+ le |119905 minus 119909|2119903+ if |119905 minus 119909| ge 1

|119905 minus 119909|119903119901minus le |119905 minus 119909|

2119903minus if |119905 minus 119909| lt 1(68)

We combine this with (16) and see that for 120582 = 119889119903119892

119901119903infin

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903+ + |119905 minus 119909|2119903minus 119889120588 (119905) 119889120588 (119909)

le 119862119901+minus1

0(119862119903+

119899minus119903+ + 119862119903minus

119899minus119903minus) le 119862

119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus

(69)

When 119899 ge 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus we take

= 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin(119862119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus)1119901+ (70)

and find

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

)

119901(119909)

119889120588 (119909) le 1 (71)

which implies

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

le le 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

(72)

Thus by Theorem 6 and taking infimum over 119892 isin 119882119903infin

119901 we

have1003817100381710038171003817119871119899(119891) minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

1003817100381710038171003817119871119899(119891 minus 119892)

1003817100381710038171003817119871119901(sdot)120588+

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

+1003817100381710038171003817119892 minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

(119872119901119887 + 1)1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588

+1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot)

(73)

where the constant 119860119901119887119889 is given by

119860119901119887119889 = 119872119901119887 + 1 + 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(74)

When 119899 lt 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus then from the

inequality 119891 minus 119892119871119901(sdot)

120588

+ 119905119892119901119903infin

ge 119905119891119871119901(sdot)

120588

valid for 119905 le 1

we observe

K119903(119891 119899minus119903minus119901+)

119901(sdot)ge 119899

minus119903minus119901+10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

(75)

and applyingTheorem 6 directly yields

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1)1003817100381710038171003817119891

1003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1) 119899119903minus119901+K119903(119891 119899

minus119903minus119901+)119901(sdot)

(76)

and thereby (21) by setting

119860119901119887119889 = (119872119901119887 + 1) 1198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(77)

The proof of Theorem 7 is complete

Abstract and Applied Analysis 9

6 Further Topics and Discussion

Approximation by linear operators is an important topicin approximation theory It mainly consists of two familiesof approximation schemes Bernstein type positive linearoperators and quasi-interpolation type linear operators inmultivariate approximation

In this paper we mainly consider Bernstein type pos-itive linear operators We verified a conjecture in [11]about the uniformboundedness of Bernstein-Durrmeyer andBernstein-Kantorovich operators with respect to an arbitraryBorel measure on (0 1) on the variable 119871

119901(sdot)

120588space under

the assumption of log-Holder continuity of the exponentfunction 119901 We also provide quantitative estimates for highorders of approximation on the variable 119871

119901(sdot)

120588by linear

combinations of Bernstein type positive linear operatorsThe study of quasi-interpolation type linear operators

started with the classical work of Schoenberg on cardinalinterpolation by B-splines It has been developed significantlydue to important applications in the areas of finite elementmethods cardinal interpolation for multivariate approxima-tion and wavelet analysis A large class of linear operators forapproximating functions on R119889 take the form

119879 (119891 119909) = intR119889

Φ (119909 119905) 119891 (119905) 119889119905 119909 isin R119889 (78)

where Φ R119889times R119889

rarr R is a window functionsatisfying int

R119889Φ(119909 119905)119889119905 = 1 and some conditions for decays

of |Φ(119909 119905) as |119909 minus 119905| increases Quantitative estimates for theapproximation of functions in 119862(R119889

) cap 119871infin

(R119889) or 119871

119901(R119889

)

with 1 le 119901 lt infin can be found in a large literature ofmultivariate approximation (see eg [20 21 28]) Establish-ing analysis for approximation by quasi-interpolation typelinear operators on the variable 119871

119901(sdot)

120588spaces would be an

interesting topic An immediate barrier we meet with suchanalysis is the boundedness assumption of the measure 120588

(120588(Ω) lt infin) This assumption is not satisfied for mostquasi-interpolation type linear operators or the classicalWeierstrass (or Gaussian convolution) operators W119899(119891) =

(radic1198992120587)119889

intR119889

expminus119899119905 minus 1199092

22119891(119905)119889119905 for which 120588 is often

the Lebesgue measure on R119889 It is desirable to overcomethe technical difficulty and establish error analysis for linearoperators with respect to unbounded measures

We describedmotivations of our study in learning theoryIt would be interesting to implement detailed error analysisfor some related learning algorithms in classification andquantile regression by means of our results on orders ofapproximation by linear operators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their constructive suggestions and comments The work

described in this paper is supported partially by the ResearchGrants Council of Hong Kong (Project no CityU 105011)Thecorresponding author is Ding-Xuan Zhou

References

[1] S N Bernstein ldquoDemonstration du teoreme de Weirerstrassfondee sur le calcul des probabilitesrdquo Communications of theKharkov Mathematical Society vol 13 pp 1ndash2 1913

[2] L V Kantorovich ldquoSur certaines developments suivant lespolynomes de la forme de S Bernstein I-IIrdquo Comptes Rendus delrsquoAcademie des Sciences de LrsquoURSS A vol 563ndash568 pp 595ndash6001930

[3] J L Durrmeyer Une formule drsquoinversion de la transformeeLaplace applications a la theorie des moments [These de 3e cycleSciences] Faculte des Sciences lrsquoUniversite Paris Paris France1967

[4] H Berens and G G Lorentz ldquoInverse theorems for Bernsteinpolynomialsrdquo Indiana University Mathematics Journal vol 21pp 693ndash708 1972

[5] H Berens and R A DeVore ldquoQuantitative Korovkin theoremsfor positive linear operators on 119871119875-spacesrdquo Transactions of theAmerican Mathematical Society vol 245 pp 349ndash361 1978

[6] Z Ditzian and V TotikModuli of Smoothness vol 9 of SpringerSeries in Computational Mathematics Springer New York NYUSA 1987

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents Springer BerlinGermany 2011

[8] W Orlicz ldquoUber konjugierte Exponentenfolgenrdquo Studia Math-ematica vol 3 pp 200ndash211 1931

[9] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[10] O Kovacik and J Rakosnk ldquoOn spaces 119871119901(119909) and 119882

1119901(119909)rdquoCzechoslovakMathematical Journal vol 41 no 116 pp 592ndash6181991

[11] D X Zhou ldquoApproximation by positive linear operators onvariables 119871

119901(119909) spacesrdquo Journal of Applied Functional Analysisvol 9 no 3-4 pp 379ndash391 2014

[12] D X Zhou and K Jetter ldquoApproximation with polynomialkernels and SVM classifiersrdquo Advances in Computational Math-ematics vol 25 no 1ndash3 pp 323ndash344 2006

[13] E E Berdysheva and K Jetter ldquoMultivariate Bernstein-Durrmeyer operators with arbitrary weight functionsrdquo Journalof Approximation Theory vol 162 no 3 pp 576ndash598 2010

[14] A B Tsybakov ldquoOptimal aggregation of classifiers in statisticallearningrdquo The Annals of Statistics vol 32 no 1 pp 135ndash1662004

[15] S Smale andD X Zhou ldquoLearning theory estimates via integraloperators and their approximationsrdquo Constructive Approxima-tion vol 26 no 2 pp 153ndash172 2007

[16] S Smale and D X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquoThe American MathematicalSociety Bulletin vol 41 no 3 pp 279ndash305 2004

[17] T Hu J Fan Q Wu and D X Zhou ldquoRegularization schemesfor minimum error entropy principlerdquo Analysis and Applica-tions 2014

[18] I Steinwart and A Christmann ldquoEstimating conditional quan-tiles with the help of the pinball lossrdquo Bernoulli vol 17 no 1 pp211ndash225 2011

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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OptimizationJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

When |119891(119905)| ge 1 we find

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119909)minus119901(119905)

le (119899120574)4119860119901 log 119899

le 1198994120574119860119901 log 119899 le 119901 (52)

where the constant number 119901 defined by

119901 = sup119899isinN

1198994120574119860119901 log 119899 (53)

is finite because

log 1198994120574119860119901 log 119899 =

4120574119860119901 log 119899

log 119899997888rarr 4120574119860119901 as 119899 997888rarr infin

(54)

When |119891(119905)| lt 1 we simply use |119891(119905)|119901(119909)minus119901(119905)

le

|119891(119905)|minus119901(119905) Applying these bounds we can estimate the core

part of Step 2 as

intΩ

intΩ119899119909

119869119899 (119891 119909 119905)1003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

le intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

(119901 +1003816100381610038161003816119891 (119905)

1003816100381610038161003816minus119901(119905)

)

times 119889120588 (119905) 119889120588 (119909)

= 119901 intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)10038161003816100381610038161003816100381610038161003816119891 (119905)

1003816100381610038161003816119901(119905)

119889120588 (119905) 119889120588 (119909)

+ intΩ

intΩ119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119889120588 (119905) 119889120588 (119909)

le 1199011198620 intΩ

1003816100381610038161003816119891 (119905)1003816100381610038161003816119901(119905)

119889120588 (119905)

+ 1198620120588 (Ω) le 1198620119901 + 1198620120588 (Ω)

(55)

Here we have used the assumption intΩ

|119891(119905)|119901(119905)

119889120588(119905) le 1 and(14) So the second term of (44) can be estimated as

intΩ

10038161003816100381610038161198711198992 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+0

(119901 + 120588 (Ω)) (56)

Step 3 Estimating the Third Term of (44) For 119905 isin Ω1015840

119899119909 we

have |119891(119905)| le 119899120574 and |119909 minus 119905| gt 119899

minus14 yielding 11989914

|119909 minus 119905| gt 1 Itfollows that

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816 le int

Ω1015840119899119909

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 119899

120574(119899

14|119909 minus 119905|)

2119903119901(119909)

119889120588 (119905)

(57)

where 119903 is the integer part of 2120574119901+ + 1 Applying the Holderinequality and (14) we see that

10038161003816100381610038161198711198993(119891 119909)1003816100381610038161003816119901(119909)

le intΩ1015840119899119909

119899120574119901(119909)+(1199032) 1003816100381610038161003816119870119899 (119909 119905)

1003816100381610038161003816 |119909 minus 119905|2119903

119889120588 (119905) 119862119901(119909)minus1

0

(58)

So by the bound 119901(119909) le 119901+ and condition (16) the third termof (44) can be bounded as

intΩ

10038161003816100381610038161198711198993 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 119862119901+minus1

0119899120574119901++(1199032)119862119903119899

minus119903

= 119862119901+minus1

0119862119903119899

120574119901+minus(1199032) le 119862119901+minus1

0119862119903

(59)

Finally we put the estimates (48) (56) and (59) into (44)to conclude

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909)

le 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(60)

Take

120582 = 119872119901119887 = 1 + 3119901+ (120588 (Ω) (119862

119901minus

119887+ 119862

119901+

119887)

+119862119901+0

(119901 + 120588 (Ω)) + 119862119901+minus1

0119862119903)

(61)

we find

intΩ

(

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le (1

120582)

119901minus

intΩ

1003816100381610038161003816119871119899 (119891 119909)1003816100381610038161003816119901(119909)

119889120588 (119909) le 1

(62)

This implies 119871119899(119891)119871119901(sdot)

120588

le 119872119901119887 The bound 119872119901119887 isindependent of 119891 So we have 119871119899 le 119872119901119887 The proofTheorem 6 is complete

We are now in a position to proveTheorem 7

Proof of Theorem 7 We follow the standard procedure inapproximation theory and consider the error 119871119899(119892 119909) minus 119892(119909)

for 119892 isin 119882119903infin

119901 Apply the Taylor expansion

119892 (119905) = 119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119909 119905 isin Ω

(63)

where the remainder term 119877119892119903(119909 119905) is given by

119877119892119903 (119909 119905)

= int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572(119905 minus 119909)

120572119889119906

(64)

8 Abstract and Applied Analysis

We see from the vanishing moment condition (20) that

119871119899 (119892 119909) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119889120588 (119905) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572

times (119905 minus 119909)120572119889119906 119889120588 (119905)

(65)

SinceΩ is convex 119909+119906(119905minus119909) isin Ω for any 119906 isin [0 1] 119909 119905 isin ΩSo |119863

120572119892(119909 + 119906(119905 minus 119909))| le 119892

119901119903infinand we have

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

le int

1

0

(1 minus 119906)119903minus1

times sum

|120572|1=119903

10038171003817100381710038171198921003817100381710038171003817119901119903infin

120572int

Ω

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

|120572|119889120588 (119905) 119889119906

le 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588 (119905)

(66)

By (14) and the Holder inequality

(intΩ

1003816100381610038161003816119870119899(119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588(119905))

119901(119909)

le intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119901(119909)119889120588 (119905) 119862

119901(119909)minus1

0

(67)

If we denote the largest integer 119903minus satisfying 2119903minus le 119903119901minus as 119903minusand the smallest integer 119903+ satisfying 2119903+ ge 119903119901+ we find

|119905 minus 119909|119903119901(119909)

le |119905 minus 119909|

119903119901+ le |119905 minus 119909|2119903+ if |119905 minus 119909| ge 1

|119905 minus 119909|119903119901minus le |119905 minus 119909|

2119903minus if |119905 minus 119909| lt 1(68)

We combine this with (16) and see that for 120582 = 119889119903119892

119901119903infin

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903+ + |119905 minus 119909|2119903minus 119889120588 (119905) 119889120588 (119909)

le 119862119901+minus1

0(119862119903+

119899minus119903+ + 119862119903minus

119899minus119903minus) le 119862

119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus

(69)

When 119899 ge 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus we take

= 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin(119862119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus)1119901+ (70)

and find

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

)

119901(119909)

119889120588 (119909) le 1 (71)

which implies

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

le le 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

(72)

Thus by Theorem 6 and taking infimum over 119892 isin 119882119903infin

119901 we

have1003817100381710038171003817119871119899(119891) minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

1003817100381710038171003817119871119899(119891 minus 119892)

1003817100381710038171003817119871119901(sdot)120588+

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

+1003817100381710038171003817119892 minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

(119872119901119887 + 1)1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588

+1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot)

(73)

where the constant 119860119901119887119889 is given by

119860119901119887119889 = 119872119901119887 + 1 + 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(74)

When 119899 lt 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus then from the

inequality 119891 minus 119892119871119901(sdot)

120588

+ 119905119892119901119903infin

ge 119905119891119871119901(sdot)

120588

valid for 119905 le 1

we observe

K119903(119891 119899minus119903minus119901+)

119901(sdot)ge 119899

minus119903minus119901+10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

(75)

and applyingTheorem 6 directly yields

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1)1003817100381710038171003817119891

1003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1) 119899119903minus119901+K119903(119891 119899

minus119903minus119901+)119901(sdot)

(76)

and thereby (21) by setting

119860119901119887119889 = (119872119901119887 + 1) 1198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(77)

The proof of Theorem 7 is complete

Abstract and Applied Analysis 9

6 Further Topics and Discussion

Approximation by linear operators is an important topicin approximation theory It mainly consists of two familiesof approximation schemes Bernstein type positive linearoperators and quasi-interpolation type linear operators inmultivariate approximation

In this paper we mainly consider Bernstein type pos-itive linear operators We verified a conjecture in [11]about the uniformboundedness of Bernstein-Durrmeyer andBernstein-Kantorovich operators with respect to an arbitraryBorel measure on (0 1) on the variable 119871

119901(sdot)

120588space under

the assumption of log-Holder continuity of the exponentfunction 119901 We also provide quantitative estimates for highorders of approximation on the variable 119871

119901(sdot)

120588by linear

combinations of Bernstein type positive linear operatorsThe study of quasi-interpolation type linear operators

started with the classical work of Schoenberg on cardinalinterpolation by B-splines It has been developed significantlydue to important applications in the areas of finite elementmethods cardinal interpolation for multivariate approxima-tion and wavelet analysis A large class of linear operators forapproximating functions on R119889 take the form

119879 (119891 119909) = intR119889

Φ (119909 119905) 119891 (119905) 119889119905 119909 isin R119889 (78)

where Φ R119889times R119889

rarr R is a window functionsatisfying int

R119889Φ(119909 119905)119889119905 = 1 and some conditions for decays

of |Φ(119909 119905) as |119909 minus 119905| increases Quantitative estimates for theapproximation of functions in 119862(R119889

) cap 119871infin

(R119889) or 119871

119901(R119889

)

with 1 le 119901 lt infin can be found in a large literature ofmultivariate approximation (see eg [20 21 28]) Establish-ing analysis for approximation by quasi-interpolation typelinear operators on the variable 119871

119901(sdot)

120588spaces would be an

interesting topic An immediate barrier we meet with suchanalysis is the boundedness assumption of the measure 120588

(120588(Ω) lt infin) This assumption is not satisfied for mostquasi-interpolation type linear operators or the classicalWeierstrass (or Gaussian convolution) operators W119899(119891) =

(radic1198992120587)119889

intR119889

expminus119899119905 minus 1199092

22119891(119905)119889119905 for which 120588 is often

the Lebesgue measure on R119889 It is desirable to overcomethe technical difficulty and establish error analysis for linearoperators with respect to unbounded measures

We describedmotivations of our study in learning theoryIt would be interesting to implement detailed error analysisfor some related learning algorithms in classification andquantile regression by means of our results on orders ofapproximation by linear operators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their constructive suggestions and comments The work

described in this paper is supported partially by the ResearchGrants Council of Hong Kong (Project no CityU 105011)Thecorresponding author is Ding-Xuan Zhou

References

[1] S N Bernstein ldquoDemonstration du teoreme de Weirerstrassfondee sur le calcul des probabilitesrdquo Communications of theKharkov Mathematical Society vol 13 pp 1ndash2 1913

[2] L V Kantorovich ldquoSur certaines developments suivant lespolynomes de la forme de S Bernstein I-IIrdquo Comptes Rendus delrsquoAcademie des Sciences de LrsquoURSS A vol 563ndash568 pp 595ndash6001930

[3] J L Durrmeyer Une formule drsquoinversion de la transformeeLaplace applications a la theorie des moments [These de 3e cycleSciences] Faculte des Sciences lrsquoUniversite Paris Paris France1967

[4] H Berens and G G Lorentz ldquoInverse theorems for Bernsteinpolynomialsrdquo Indiana University Mathematics Journal vol 21pp 693ndash708 1972

[5] H Berens and R A DeVore ldquoQuantitative Korovkin theoremsfor positive linear operators on 119871119875-spacesrdquo Transactions of theAmerican Mathematical Society vol 245 pp 349ndash361 1978

[6] Z Ditzian and V TotikModuli of Smoothness vol 9 of SpringerSeries in Computational Mathematics Springer New York NYUSA 1987

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents Springer BerlinGermany 2011

[8] W Orlicz ldquoUber konjugierte Exponentenfolgenrdquo Studia Math-ematica vol 3 pp 200ndash211 1931

[9] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[10] O Kovacik and J Rakosnk ldquoOn spaces 119871119901(119909) and 119882

1119901(119909)rdquoCzechoslovakMathematical Journal vol 41 no 116 pp 592ndash6181991

[11] D X Zhou ldquoApproximation by positive linear operators onvariables 119871

119901(119909) spacesrdquo Journal of Applied Functional Analysisvol 9 no 3-4 pp 379ndash391 2014

[12] D X Zhou and K Jetter ldquoApproximation with polynomialkernels and SVM classifiersrdquo Advances in Computational Math-ematics vol 25 no 1ndash3 pp 323ndash344 2006

[13] E E Berdysheva and K Jetter ldquoMultivariate Bernstein-Durrmeyer operators with arbitrary weight functionsrdquo Journalof Approximation Theory vol 162 no 3 pp 576ndash598 2010

[14] A B Tsybakov ldquoOptimal aggregation of classifiers in statisticallearningrdquo The Annals of Statistics vol 32 no 1 pp 135ndash1662004

[15] S Smale andD X Zhou ldquoLearning theory estimates via integraloperators and their approximationsrdquo Constructive Approxima-tion vol 26 no 2 pp 153ndash172 2007

[16] S Smale and D X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquoThe American MathematicalSociety Bulletin vol 41 no 3 pp 279ndash305 2004

[17] T Hu J Fan Q Wu and D X Zhou ldquoRegularization schemesfor minimum error entropy principlerdquo Analysis and Applica-tions 2014

[18] I Steinwart and A Christmann ldquoEstimating conditional quan-tiles with the help of the pinball lossrdquo Bernoulli vol 17 no 1 pp211ndash225 2011

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

We see from the vanishing moment condition (20) that

119871119899 (119892 119909) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

119892 (119909) + sum

1le|120572|1le119903minus1

119863120572119892 (119909)

120572(119905 minus 119909)

120572

+ 119877119892119903 (119909 119905) 119889120588 (119905) minus 119892 (119909)

= intΩ

119870119899 (119909 119905)

int

1

0

(1 minus 119906)119903minus1

sum

|120572|1=119903

119863120572119892 (119909 + 119906 (119905 minus 119909))

120572

times (119905 minus 119909)120572119889119906 119889120588 (119905)

(65)

SinceΩ is convex 119909+119906(119905minus119909) isin Ω for any 119906 isin [0 1] 119909 119905 isin ΩSo |119863

120572119892(119909 + 119906(119905 minus 119909))| le 119892

119901119903infinand we have

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

le int

1

0

(1 minus 119906)119903minus1

times sum

|120572|1=119903

10038171003817100381710038171198921003817100381710038171003817119901119903infin

120572int

Ω

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

|120572|119889120588 (119905) 119889119906

le 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588 (119905)

(66)

By (14) and the Holder inequality

(intΩ

1003816100381610038161003816119870119899(119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119889120588(119905))

119901(119909)

le intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

119903119901(119909)119889120588 (119905) 119862

119901(119909)minus1

0

(67)

If we denote the largest integer 119903minus satisfying 2119903minus le 119903119901minus as 119903minusand the smallest integer 119903+ satisfying 2119903+ ge 119903119901+ we find

|119905 minus 119909|119903119901(119909)

le |119905 minus 119909|

119903119901+ le |119905 minus 119909|2119903+ if |119905 minus 119909| ge 1

|119905 minus 119909|119903119901minus le |119905 minus 119909|

2119903minus if |119905 minus 119909| lt 1(68)

We combine this with (16) and see that for 120582 = 119889119903119892

119901119903infin

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

120582)

119901(119909)

119889120588 (119909)

le 119862119901+minus1

0intΩ

intΩ

1003816100381610038161003816119870119899 (119909 119905)1003816100381610038161003816 |119905 minus 119909|

2119903+ + |119905 minus 119909|2119903minus 119889120588 (119905) 119889120588 (119909)

le 119862119901+minus1

0(119862119903+

119899minus119903+ + 119862119903minus

119899minus119903minus) le 119862

119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus

(69)

When 119899 ge 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus we take

= 1198891199031003817100381710038171003817119892

1003817100381710038171003817119901119903infin(119862119901+minus1

0(119862119903+

+ 119862119903minus) 119899

minus119903minus)1119901+ (70)

and find

intΩ

(

1003816100381610038161003816119871119899 (119892 119909) minus 119892 (119909)1003816100381610038161003816

)

119901(119909)

119889120588 (119909) le 1 (71)

which implies

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

le le 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

(72)

Thus by Theorem 6 and taking infimum over 119892 isin 119882119903infin

119901 we

have1003817100381710038171003817119871119899(119891) minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

1003817100381710038171003817119871119899(119891 minus 119892)

1003817100381710038171003817119871119901(sdot)120588+

1003817100381710038171003817119871119899(119892) minus 1198921003817100381710038171003817119871119901(sdot)120588

+1003817100381710038171003817119892 minus 119891

1003817100381710038171003817119871119901(sdot)120588

le inf119892isin119882119903infin119901

(119872119901119887 + 1)1003817100381710038171003817119891 minus 119892

1003817100381710038171003817119871119901(sdot)120588

+1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

119899minus119903minus119901+1003817100381710038171003817119892

1003817100381710038171003817119901119903infin

le 119860119901119887119889K119903(119891 119899minus119903minus119901+)

119901(sdot)

(73)

where the constant 119860119901119887119889 is given by

119860119901119887119889 = 119872119901119887 + 1 + 1198891199031198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(74)

When 119899 lt 119862(119901+minus1)119903minus0

(119862119903++ 119862119903minus

)1119903minus then from the

inequality 119891 minus 119892119871119901(sdot)

120588

+ 119905119892119901119903infin

ge 119905119891119871119901(sdot)

120588

valid for 119905 le 1

we observe

K119903(119891 119899minus119903minus119901+)

119901(sdot)ge 119899

minus119903minus119901+10038171003817100381710038171198911003817100381710038171003817119871119901(sdot)120588

(75)

and applyingTheorem 6 directly yields

1003817100381710038171003817119871119899(119891) minus 1198911003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1)1003817100381710038171003817119891

1003817100381710038171003817119871119901(sdot)120588

le (119872119901119887 + 1) 119899119903minus119901+K119903(119891 119899

minus119903minus119901+)119901(sdot)

(76)

and thereby (21) by setting

119860119901119887119889 = (119872119901119887 + 1) 1198621minus(1119901+)

0(119862119903+

+ 119862119903minus)1119901+

(77)

The proof of Theorem 7 is complete

Abstract and Applied Analysis 9

6 Further Topics and Discussion

Approximation by linear operators is an important topicin approximation theory It mainly consists of two familiesof approximation schemes Bernstein type positive linearoperators and quasi-interpolation type linear operators inmultivariate approximation

In this paper we mainly consider Bernstein type pos-itive linear operators We verified a conjecture in [11]about the uniformboundedness of Bernstein-Durrmeyer andBernstein-Kantorovich operators with respect to an arbitraryBorel measure on (0 1) on the variable 119871

119901(sdot)

120588space under

the assumption of log-Holder continuity of the exponentfunction 119901 We also provide quantitative estimates for highorders of approximation on the variable 119871

119901(sdot)

120588by linear

combinations of Bernstein type positive linear operatorsThe study of quasi-interpolation type linear operators

started with the classical work of Schoenberg on cardinalinterpolation by B-splines It has been developed significantlydue to important applications in the areas of finite elementmethods cardinal interpolation for multivariate approxima-tion and wavelet analysis A large class of linear operators forapproximating functions on R119889 take the form

119879 (119891 119909) = intR119889

Φ (119909 119905) 119891 (119905) 119889119905 119909 isin R119889 (78)

where Φ R119889times R119889

rarr R is a window functionsatisfying int

R119889Φ(119909 119905)119889119905 = 1 and some conditions for decays

of |Φ(119909 119905) as |119909 minus 119905| increases Quantitative estimates for theapproximation of functions in 119862(R119889

) cap 119871infin

(R119889) or 119871

119901(R119889

)

with 1 le 119901 lt infin can be found in a large literature ofmultivariate approximation (see eg [20 21 28]) Establish-ing analysis for approximation by quasi-interpolation typelinear operators on the variable 119871

119901(sdot)

120588spaces would be an

interesting topic An immediate barrier we meet with suchanalysis is the boundedness assumption of the measure 120588

(120588(Ω) lt infin) This assumption is not satisfied for mostquasi-interpolation type linear operators or the classicalWeierstrass (or Gaussian convolution) operators W119899(119891) =

(radic1198992120587)119889

intR119889

expminus119899119905 minus 1199092

22119891(119905)119889119905 for which 120588 is often

the Lebesgue measure on R119889 It is desirable to overcomethe technical difficulty and establish error analysis for linearoperators with respect to unbounded measures

We describedmotivations of our study in learning theoryIt would be interesting to implement detailed error analysisfor some related learning algorithms in classification andquantile regression by means of our results on orders ofapproximation by linear operators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their constructive suggestions and comments The work

described in this paper is supported partially by the ResearchGrants Council of Hong Kong (Project no CityU 105011)Thecorresponding author is Ding-Xuan Zhou

References

[1] S N Bernstein ldquoDemonstration du teoreme de Weirerstrassfondee sur le calcul des probabilitesrdquo Communications of theKharkov Mathematical Society vol 13 pp 1ndash2 1913

[2] L V Kantorovich ldquoSur certaines developments suivant lespolynomes de la forme de S Bernstein I-IIrdquo Comptes Rendus delrsquoAcademie des Sciences de LrsquoURSS A vol 563ndash568 pp 595ndash6001930

[3] J L Durrmeyer Une formule drsquoinversion de la transformeeLaplace applications a la theorie des moments [These de 3e cycleSciences] Faculte des Sciences lrsquoUniversite Paris Paris France1967

[4] H Berens and G G Lorentz ldquoInverse theorems for Bernsteinpolynomialsrdquo Indiana University Mathematics Journal vol 21pp 693ndash708 1972

[5] H Berens and R A DeVore ldquoQuantitative Korovkin theoremsfor positive linear operators on 119871119875-spacesrdquo Transactions of theAmerican Mathematical Society vol 245 pp 349ndash361 1978

[6] Z Ditzian and V TotikModuli of Smoothness vol 9 of SpringerSeries in Computational Mathematics Springer New York NYUSA 1987

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents Springer BerlinGermany 2011

[8] W Orlicz ldquoUber konjugierte Exponentenfolgenrdquo Studia Math-ematica vol 3 pp 200ndash211 1931

[9] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[10] O Kovacik and J Rakosnk ldquoOn spaces 119871119901(119909) and 119882

1119901(119909)rdquoCzechoslovakMathematical Journal vol 41 no 116 pp 592ndash6181991

[11] D X Zhou ldquoApproximation by positive linear operators onvariables 119871

119901(119909) spacesrdquo Journal of Applied Functional Analysisvol 9 no 3-4 pp 379ndash391 2014

[12] D X Zhou and K Jetter ldquoApproximation with polynomialkernels and SVM classifiersrdquo Advances in Computational Math-ematics vol 25 no 1ndash3 pp 323ndash344 2006

[13] E E Berdysheva and K Jetter ldquoMultivariate Bernstein-Durrmeyer operators with arbitrary weight functionsrdquo Journalof Approximation Theory vol 162 no 3 pp 576ndash598 2010

[14] A B Tsybakov ldquoOptimal aggregation of classifiers in statisticallearningrdquo The Annals of Statistics vol 32 no 1 pp 135ndash1662004

[15] S Smale andD X Zhou ldquoLearning theory estimates via integraloperators and their approximationsrdquo Constructive Approxima-tion vol 26 no 2 pp 153ndash172 2007

[16] S Smale and D X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquoThe American MathematicalSociety Bulletin vol 41 no 3 pp 279ndash305 2004

[17] T Hu J Fan Q Wu and D X Zhou ldquoRegularization schemesfor minimum error entropy principlerdquo Analysis and Applica-tions 2014

[18] I Steinwart and A Christmann ldquoEstimating conditional quan-tiles with the help of the pinball lossrdquo Bernoulli vol 17 no 1 pp211ndash225 2011

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

6 Further Topics and Discussion

Approximation by linear operators is an important topicin approximation theory It mainly consists of two familiesof approximation schemes Bernstein type positive linearoperators and quasi-interpolation type linear operators inmultivariate approximation

In this paper we mainly consider Bernstein type pos-itive linear operators We verified a conjecture in [11]about the uniformboundedness of Bernstein-Durrmeyer andBernstein-Kantorovich operators with respect to an arbitraryBorel measure on (0 1) on the variable 119871

119901(sdot)

120588space under

the assumption of log-Holder continuity of the exponentfunction 119901 We also provide quantitative estimates for highorders of approximation on the variable 119871

119901(sdot)

120588by linear

combinations of Bernstein type positive linear operatorsThe study of quasi-interpolation type linear operators

started with the classical work of Schoenberg on cardinalinterpolation by B-splines It has been developed significantlydue to important applications in the areas of finite elementmethods cardinal interpolation for multivariate approxima-tion and wavelet analysis A large class of linear operators forapproximating functions on R119889 take the form

119879 (119891 119909) = intR119889

Φ (119909 119905) 119891 (119905) 119889119905 119909 isin R119889 (78)

where Φ R119889times R119889

rarr R is a window functionsatisfying int

R119889Φ(119909 119905)119889119905 = 1 and some conditions for decays

of |Φ(119909 119905) as |119909 minus 119905| increases Quantitative estimates for theapproximation of functions in 119862(R119889

) cap 119871infin

(R119889) or 119871

119901(R119889

)

with 1 le 119901 lt infin can be found in a large literature ofmultivariate approximation (see eg [20 21 28]) Establish-ing analysis for approximation by quasi-interpolation typelinear operators on the variable 119871

119901(sdot)

120588spaces would be an

interesting topic An immediate barrier we meet with suchanalysis is the boundedness assumption of the measure 120588

(120588(Ω) lt infin) This assumption is not satisfied for mostquasi-interpolation type linear operators or the classicalWeierstrass (or Gaussian convolution) operators W119899(119891) =

(radic1198992120587)119889

intR119889

expminus119899119905 minus 1199092

22119891(119905)119889119905 for which 120588 is often

the Lebesgue measure on R119889 It is desirable to overcomethe technical difficulty and establish error analysis for linearoperators with respect to unbounded measures

We describedmotivations of our study in learning theoryIt would be interesting to implement detailed error analysisfor some related learning algorithms in classification andquantile regression by means of our results on orders ofapproximation by linear operators

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the anonymous refereesfor their constructive suggestions and comments The work

described in this paper is supported partially by the ResearchGrants Council of Hong Kong (Project no CityU 105011)Thecorresponding author is Ding-Xuan Zhou

References

[1] S N Bernstein ldquoDemonstration du teoreme de Weirerstrassfondee sur le calcul des probabilitesrdquo Communications of theKharkov Mathematical Society vol 13 pp 1ndash2 1913

[2] L V Kantorovich ldquoSur certaines developments suivant lespolynomes de la forme de S Bernstein I-IIrdquo Comptes Rendus delrsquoAcademie des Sciences de LrsquoURSS A vol 563ndash568 pp 595ndash6001930

[3] J L Durrmeyer Une formule drsquoinversion de la transformeeLaplace applications a la theorie des moments [These de 3e cycleSciences] Faculte des Sciences lrsquoUniversite Paris Paris France1967

[4] H Berens and G G Lorentz ldquoInverse theorems for Bernsteinpolynomialsrdquo Indiana University Mathematics Journal vol 21pp 693ndash708 1972

[5] H Berens and R A DeVore ldquoQuantitative Korovkin theoremsfor positive linear operators on 119871119875-spacesrdquo Transactions of theAmerican Mathematical Society vol 245 pp 349ndash361 1978

[6] Z Ditzian and V TotikModuli of Smoothness vol 9 of SpringerSeries in Computational Mathematics Springer New York NYUSA 1987

[7] L Diening P Harjulehto P Hasto and M Ruzicka Lebesgueand Sobolev Spaces with Variable Exponents Springer BerlinGermany 2011

[8] W Orlicz ldquoUber konjugierte Exponentenfolgenrdquo Studia Math-ematica vol 3 pp 200ndash211 1931

[9] E Acerbi and G Mingione ldquoRegularity results for a class offunctionals with non-standard growthrdquo Archive for RationalMechanics and Analysis vol 156 no 2 pp 121ndash140 2001

[10] O Kovacik and J Rakosnk ldquoOn spaces 119871119901(119909) and 119882

1119901(119909)rdquoCzechoslovakMathematical Journal vol 41 no 116 pp 592ndash6181991

[11] D X Zhou ldquoApproximation by positive linear operators onvariables 119871

119901(119909) spacesrdquo Journal of Applied Functional Analysisvol 9 no 3-4 pp 379ndash391 2014

[12] D X Zhou and K Jetter ldquoApproximation with polynomialkernels and SVM classifiersrdquo Advances in Computational Math-ematics vol 25 no 1ndash3 pp 323ndash344 2006

[13] E E Berdysheva and K Jetter ldquoMultivariate Bernstein-Durrmeyer operators with arbitrary weight functionsrdquo Journalof Approximation Theory vol 162 no 3 pp 576ndash598 2010

[14] A B Tsybakov ldquoOptimal aggregation of classifiers in statisticallearningrdquo The Annals of Statistics vol 32 no 1 pp 135ndash1662004

[15] S Smale andD X Zhou ldquoLearning theory estimates via integraloperators and their approximationsrdquo Constructive Approxima-tion vol 26 no 2 pp 153ndash172 2007

[16] S Smale and D X Zhou ldquoShannon sampling and functionreconstruction from point valuesrdquoThe American MathematicalSociety Bulletin vol 41 no 3 pp 279ndash305 2004

[17] T Hu J Fan Q Wu and D X Zhou ldquoRegularization schemesfor minimum error entropy principlerdquo Analysis and Applica-tions 2014

[18] I Steinwart and A Christmann ldquoEstimating conditional quan-tiles with the help of the pinball lossrdquo Bernoulli vol 17 no 1 pp211ndash225 2011

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[19] D H Xiang ldquoA new comparison theorem on conditionalquantilesrdquoAppliedMathematics Letters vol 25 no 1 pp 58ndash622012

[20] J Lei R Jia and E W Cheney ldquoApproximation from shift-invariant spaces by integral operatorsrdquo SIAM Journal on Math-ematical Analysis vol 28 no 2 pp 481ndash498 1997

[21] K Jetter and D X Zhou ldquoOrder of linear approximation fromshift-invariant spacesrdquo Constructive Approximation vol 11 no4 pp 423ndash438 1995

[22] M Derriennic ldquoOn multivariate approximation by Bernstein-type polynomialsrdquo Journal of ApproximationTheory vol 45 no2 pp 155ndash166 1985

[23] H Berens and Y Xu ldquoOn Bernstein-Durrmeyer polynomialswith Jacobi weightsrdquo in Approximation Theory and FunctionalAnalysis C K Chui Ed pp 25ndash46 Academic Press BostonMass USA 1991

[24] B-Z Li ldquoApproximation by multivariate Bernstein-Durrmeyeroperators and learning rates of least-squares regularized regres-sion with multivariate polynomial kernelsrdquo Journal of Approxi-mation Theory vol 173 pp 33ndash55 2013

[25] E E Berdysheva ldquoUniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measurerdquoJournal of Mathematical Analysis and Applications vol 394 no1 pp 324ndash336 2012

[26] E E Berdysheva ldquoBernsteinndashDurrmeyer operators withrespect to arbitrary measure II pointwise convergencerdquoJournal of Mathematical Analysis and Applications vol 418 no2 pp 734ndash752 2014

[27] D X Zhou ldquoConverse theorems for multidimensional Kan-torovich operatorsrdquoAnalysis Mathematica vol 19 no 1 pp 85ndash100 1993

[28] C de Boor R A DeVore and A Ron ldquoApproximation fromshift-invariant subspaces of 1198712(R

119889)rdquo Transactions of the Ameri-

can Mathematical Society vol 341 no 2 pp 787ndash806 1994

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of