Rate of convergence in $$L_{p}$$ L p approximation

$Page 1: Rate of convergence in $$L_{p}$$ L p approximation$
Period Math Hung (2014) 68:176–184DOI 10.1007/s10998-014-0028-1

Rate of convergence in L p approximation

C. Orhan · I. Sakaoglu

Published online: 20 May 2014© Akadémiai Kiadó, Budapest, Hungary 2014

Abstract In the present paper we give a Korovkin type approximation theorem for a sequenceof positive linear operators acting from L p [a, b] into itself using the concept of A-summationprocesses. We also study the rate of convergence of these operators.

Keywords A-summation process · Positive linear operator · Korovkin type theorem ·Second-order modulus of smoothness · Rate of convergence

Mathematics Subject Classification Primary 41A25 · 41A36 · Secondary 40A05

1 Introduction

Korovkin theorems provide conditions for whether a given sequence of positive linear oper-ators converges to the identity operator in the space of continuous functions on a compactinterval (see, [9]). Some results concerning Korovkin type approximation theorems in thespace L p [a, b] of Lebesgue integrable functions on a compact interval may be found in [5].Quantitative Korovkin theorems for approximation by positive linear operators in L p [a, b]spaces have also been studied in [3,4,16,17].

In order to get stronger results it is useful to use certain summation processes.We first recall some basic concepts used throughout the paper.Let L p [a, b] , 1 ≤ p < ∞, denote the space of measurable real valued pth power

Lebesgue integrable functions f on [a, b] with ‖ f ‖p =(∫ b

a | f |p dμ)1/p

.

C. Orhan · I. SakaogluDepartment of Mathematics, Faculty of Science, Ankara University, Tandogan, Ankara 06100, Turkeye-mail: [email protected]

Present Address:I. Sakaoglu (B)Department of Mathematics, Faculty of Science, Ankara University, Tandogan, Ankara 06100, Turkeye-mail: [email protected]

123

Rate of convergence 177

Let T : L p → L p be a linear operator. Recall that T is a positive operator if T f ≥ 0whenever f ≥ 0. If T is a positive linear operator then f ≤ g implies that T f ≤ T g, and| f | ≤ g implies that |T f | ≤ T g. The operator norm ‖T ‖L p→L p

is given by ‖T ‖L p→L p=

sup‖ f ‖p=1 ‖T f ‖p .

Let A := {A(n)

} ={

a(n)k j

}be a sequence of infinite matrices with nonnegative real

entries. A sequence{Tj

}of positive linear operators from L p [a, b] into itself is called a

strong A- summation process in L p [a, b] if{Tj f

}is strongly A-summable to f for every

f ∈ L p [a, b] , i.e.,

limk

∑j

a(n)k j

∥∥Tj f − f∥∥

p = 0, uniformly in n.

Some results concerning strong summation processes in L p [a, b] may be found in [14].A sequence

{Tj

}of positive linear operators from L p [a, b] into itself is called an A-

summation process in L p [a, b] if{Tj f

}is A-summable to f for every f ∈ L p [a, b] ,

i.e.,

limk

∥∥∥∥∥∥∑

j

a(n)k j Tj f − f

∥∥∥∥∥∥p

= 0, uniformly in n,

where it is assumed that the series converges for each k, n and f. Recall that a sequence ofreal numbers

{x j

}is said to be A-summable to L if limk

∑j a(n)k j x j = L , uniformly in n (see

[2,15]). Results concerning summation processes on other spaces may be found in [1,8,11]and [12].

The next result establishes a relationship between strong summation processes and sum-mation processes in L p [a, b] .

Proposition 1.1 Let A := {A(n)} be a sequence of infinite matrices with nonnegative realentries and assume that

limk

supn

∑j

a(n)k j = 1. (1.1)

Let {Tj } be a sequence of positive linear operators from L p[a, b], 1 ≤ p < ∞, into itself. If{Tj } is a strong A -summation process in L p[a, b] then {Tj } is an A -summation process inL p[a, b].Proof Let f ∈ L p[a, b] and {Tj } be a strong A-summation process in L p[a, b].∥∥∥∥∥∥∑

j

a(n)k j Tj ( f ; x)− f (x)

∥∥∥∥∥∥p

=∥∥∥∥∥∥∑

j

a(n)k j (Tj ( f ; x)− f (x))+ f (x)

⎛⎝∑

j

a(n)k j − 1

⎞⎠

∥∥∥∥∥∥p

≤∑

j

a(n)k j

∥∥Tj ( f ; x)− f (x)∥∥

p + ‖ f ‖p

∣∣∣∣∣∣∑

j

a(n)k j − 1

∣∣∣∣∣∣.

Then, by (1.1), we have for any function f ∈ L p[a, b], that

limk

supn

∥∥∥∥∥∥∑

j

a(n)k j Tj ( f ; x)− f (x)

∥∥∥∥∥∥p

= 0,

i.e., {Tj } is an A-summation process in L p[a, b], which concludes the proof. �

123

178 C. Orhan, I. Sakaoglu

Let {Tj } be a sequence of positive linear operators from L p[a, b], 1 ≤ p < ∞, into itself.Furthermore for each k, n ∈ N and f ∈ L p[a, b], let

B(n)k ( f ; x) :=∑

j

a(n)k j Tj ( f ; x).

The main aim of the present work is to study a Korovkin type L p approximation theoremfor a sequence of positive linear operators using a matrix summability method which includesboth convergence and almost convergence. We also approximate the rate of convergence inL p using the second-order modulus of smoothness and Peetre K -functional.

2 A-Summation processes in L p[a, b]

In this section we give a summary of Korovkin type L p approximation theorems and then,using an A-summation process, we study a Korovkin type L p approximation theorem fora sequence of positive linear operators. We also give an example of a sequence of positivelinear operators which satisfies Theorem 2.2 but does not satisfy Theorem 2.1.

Theorem 2.1 Let {Tn} be a uniformly bounded sequence of positive linear operators fromL p[a, b], 1 ≤ p < ∞, into itself. Then the sequence {Tn f } converges to f in L p norm forany function f ∈ L p[a, b] if and only if

limn

‖Tn( fi ; x)− fi (x)‖p = 0 f or i = 0, 1, 2, (2.1)

where fi (y) = yi for i = 0, 1, 2, (see [5]).

Using a summation process we get a stronger result than Theorem 2.1.

Theorem 2.2 Let A := {A(n)} be a sequence of infinite matrices with nonnegative realentries. Let {Tj } be a sequence of positive linear operators from L p[a, b], 1 ≤ p < ∞,

into itself and assume that

H := supn,k

∑j

a(n)k j

∥∥Tj∥∥

L p→L p< ∞. (2.2)

Then {Tj } is an A-summation process in L p[a, b], i.e., for any function f ∈ L p[a, b]

limk

supn

∥∥∥B(n)k ( f ; x)− f (x)∥∥∥

p= 0 (2.3)

if and only if

limk

supn

∥∥∥B(n)k ( fi ; x)− fi (x)∥∥∥

p= 0 f or i = 0, 1, 2, (2.4)

where fi (y) = yi for i = 0, 1, 2.

Proof The implication (2.3) ⇒ (2.4) is clear. Now assume that (2.4) holds. Let f ∈ L p[a, b].Since C[a, b], the set of all continuous functions on [a, b], is dense in L p[a, b], given ε > 0there exists ϕ ∈ C[a, b] such that

‖ f − ϕ‖p < ε. (2.5)

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Using the positivity and linearity of operators Tj and inequality (2.5), we obtain∥∥∥B(n)k ( f ; x)− f (x)

∥∥∥p

≤∥∥∥B(n)k ( f − ϕ; x)

∥∥∥p

+∥∥∥B(n)k (ϕ; x)− ϕ(x)

∥∥∥p

+ ‖ f − ϕ‖p

< ε

⎛⎝1+

∞∑j=1

a(n)k j

∥∥Tj∥∥

L p→L p

⎞⎠+

∥∥∥B(n)k (ϕ; x)− ϕ(x)∥∥∥

p. (2.6)

By the continuity of ϕ on [a, b], given ε > 0 there exists a number δ > 0 such that for allx, t ∈ [a, b] satisfying |t − x | < δ we have

|ϕ(t)− ϕ(x)| < ε. (2.7)

Also, for all x, t ∈ [a, b] satisfying |t − x | ≥ δ, we get that

|ϕ(t)− ϕ(x)| ≤ |ϕ(t)| + |ϕ(x)| ≤ 2M

δ2 ψ(t), (2.8)

where ψ(t) = (t − x)2 and M := ‖ϕ‖C[a,b] .Combining (2.7) with (2.8), we have

|ϕ(t)− ϕ(x)| < ε + 2M

δ2 ψ(t). (2.9)

for all x, t ∈ [a, b].Now using the fact that ϕ ∈ C[a, b] and considering the monotonicity of operators Tj ,

the second term on the right hand side of (2.6) may be written as follows:∥∥∥B(n)k (ϕ; x)− ϕ(x)

∥∥∥p

≤∥∥∥B(n)k (|ϕ(t)− ϕ(x)| ; x)

∥∥∥p

+ M∥∥∥B(n)k ( f0; x)− f0(x)

∥∥∥p. (2.10)

Then, by the monotonicity and linearity of operators Tj and (2.9) we have

∥∥∥B(n)k (|ϕ(t)− ϕ(x)| ; x)∥∥∥

p<

∥∥∥∥B(n)k (ε + 2M

δ2 ψ(t); x)

∥∥∥∥p

≤ ε + (ε + 2M

δ2 β2)

∥∥∥B(n)k ( f0; x)− f0(x)∥∥∥

p

+ 4M

δ2 β

∥∥∥B(n)k ( f1; x)− f1(x)∥∥∥

p

+ 2M

δ2

∥∥∥B(n)k ( f2; x)− f2(x)∥∥∥

p, (2.11)

where β := max{|a| , |b|}. In this case, by (2.2), (2.6), (2.10) and (2.11), we have∥∥∥B(n)k ( f ; x)− f (x)

∥∥∥p< ε(2 + H)+ 2M

δ2

∥∥∥B(n)k ( f2; x)− f2(x)∥∥∥

p

+ 4M

δ2 β

∥∥∥B(n)k ( f1; x)− f1(x)∥∥∥

p

+ (ε + M + 2M

δ2 β2)

∥∥∥B(n)k ( f0; x)− f0(x)∥∥∥

p. (2.12)

for all n, k.

123


Since ε > 0 is arbitrary, (2.4) and (2.12) yield that

limk

supn

∥∥∥B(n)k ( f ; x)− f (x)∥∥∥

p= 0,

which concludes the proof. �Remark 2.3 We now give an example of a sequence of positive linear operators which satisfiesTheorem 2.2 but does not satisfy Theorem 2.1. Assume that A := {A(n)} = {a(n)k j } is asequence of infinite matrices defined by

a(n)k j ={

1k+1 , n ≤ j ≤ n + k

0, otherwise.

In this case the A-summability method reduces to almost convergence [10].Let Tj : L p[−1, 1] → L p[−1, 1] be given by

Tj ( f ; x) ={

f (x), |x | ≤ 1 − 1j ,

f (−x), 1 − 1j < |x | ≤ 1. j = 1, 2, . . .

The sequence {Tj } is considered in [4]. It is easy to see that {Tj } satisfiesTheorem 2.1. A simple calculation shows that for all j ∈ N,

∥∥Tj∥∥

L p→L p= 1. Hence {Tj }

is a uniformly bounded sequence of positive linear operators from L p[−1, 1], 1 ≤ p < ∞,

into itself. Also (2.1) is satisfied. Now define {Pj } by

Pj ( f ; x) = (1 + u j )Tj ( f ; x)

where u j = (−1) j , j ∈ N. It is easy to see that{u j

}is almost convergent to zero [10], i.e.,

limk

1

k + 1

k+n∑j=n

u j = 0, uniformly in n.

Therefore the sequence of positive linear operators {Pj } satisfies Theorem 2.2 but does notsatisfy Theorem 2.1.

3 Rate of convergence for an A-summation process in L p[a, b]

Regarding the rate of convergence in L p[a, b] approximation, consult [3,4,16,17]. In thissection we study the rate of convergence of the summation process considered in Theorem2.2. The following Lemmas are needed to prove our main Theorem 3.4.

First, we give some basic definitions and notations used in this section.Let L(2)p [a, b] denote the space of those functions f ∈ L p[a, b] with f

′absolutely

continuous and f′′ ∈ L p[a, b] where f

′and f

′′are respectively the first and second derivatives

of f .For f ∈ L p[a, b], 1 ≤ p < ∞, and t > 0, the K -functional of Peetre (see [13]) is

defined by,

K2,p( f ; t) = infg∈L(2)p [a,b]

{‖ f − g‖p + t (‖g‖p +∥∥∥g

′′∥∥∥p)}. (3.1)

Following [3,4], for f ∈ L p[a, b], 1 ≤ p < ∞, we denote its modulus of continuity byw1,p( f ), and its second-order modulus of smoothness by w2,p( f ); the latter being definedby

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w2,p( f, h) = sup0<t≤h

‖ f (x + t)− 2 f (x)+ f (x − t)‖L p[a+t,b−t] ,

where [a + t, b − t] ⊂ [a, b].By a well known result on the modulus of smoothness, [7], we have

C−11

(min(1, t2) ‖ f ‖p + w2,p( f ; t)

) ≤ K2,p( f ; t2)

≤ C1(min(1, t2) ‖ f ‖p + w2,p( f ; t)

)(3.2)

where C1 > 0 is an absolute constant and independent of f and p.

Let μkp :=(

maxi

{sup

n

∥∥∥B(n)k (gi ; x)− gi (x)∥∥∥

p

})p/2p+1

, where gi (y) = yi for i =0, 1, 2 and

{Tj

}is a sequence of positive linear operators from L p[a, b] into itself. In this

section we assume that for a fixed p, μkp → 0 as k → ∞.

Lemma 3.1 Let g ∈ L(2)p [a, b], 1 ≤ p < ∞, and fix δ > 0. For x, t ∈ [a, b] we have∣∣∣∣∣∣

t∫

x

(t − u)g′′(u)du

∣∣∣∣∣∣≤ δ

δ∫

0

∣∣∣g′′(x + u)

∣∣∣ du + (t − x)2

δ1/p

∥∥∥g′′∥∥∥

p,

(see [17] page 85).

Lemma 3.2 Let A := {A(n)} be a sequence of infinite matrices with nonnegative real entries.Let {Tj } be a sequence of positive linear operators from L p[a, b], 1 ≤ p < ∞, into itself

and assume that (2.2) holds. Then for all k sufficently large and for all g ∈ L(2)p [a, b] wehave

supn

∥∥∥∥∥∥B(n)k

⎛⎝

t∫

x

(t − u)g′′(u)du; x

⎞⎠

∥∥∥∥∥∥p

≤ C∥∥∥g

′′∥∥∥pμ2

kp,

where C > 0 is a constant.

Proof Let g ∈ L(2)p [a, b].Using the monotonicity of operators Tj and Lemma 3.1, we obtain∥∥∥∥∥∥

B(n)k

⎛⎝

t∫

x

(t − u)g′′(u)du; x

⎞⎠

∥∥∥∥∥∥p

≤∥∥∥∥∥∥δ

⎛⎝

δ∫

0

∣∣∣g′′(x + u)

∣∣∣ du

⎞⎠ B(n)k (g0; x)

∥∥∥∥∥∥p

+

∥∥∥g′′∥∥∥

p

δ1/p

∥∥∥B(n)k ((t − x)2 ; x)∥∥∥

p. (3.3)

By the Hölder inequality and the generalized Minkowski inequality (see [18]), and the factthat g

′′(x) = 0 if x /∈ [a, b], the first term on the right hand side of (3.3) is dominated by

∥∥∥∥∥∥δ

δ∫

0

∣∣∣g′′(x + u)

∣∣∣ du(

B(n)k (g0; x)− g0(x))

+ δ

δ∫

0

∣∣∣g′′(x + u)

∣∣∣ du

∥∥∥∥∥∥p

≤ δ2−1/p∥∥∥g

′′∥∥∥p

∥∥∥B(n)k (g0; x)− g0(x)∥∥∥

p+ δ

δ∫

0

∥∥∥g′′(x + u)

∥∥∥p

du

≤∥∥∥g

′′∥∥∥p{δ2−1/p

∥∥∥B(n)k (g0; x)− g0(x)∥∥∥

p+ δ2}.

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Therefore∥∥∥∥∥∥

B(n)k

⎛⎝

t∫

x

(t − u)g′′(u)du; x

⎞⎠

∥∥∥∥∥∥p

≤∥∥∥g

′′∥∥∥p

{δ2 + 1

δ1/p

[∥∥∥B(n)k (g2; x)− g2(x)∥∥∥

p

+2β∥∥∥B(n)k (g1; x)− g1(x)

∥∥∥p

+(δ2 + β2)

∥∥∥B(n)k (g0; x)− g0(x)∥∥∥

p

]}.

For k sufficiently large we can choose δ := μkp to obtain

supn

∥∥∥∥∥∥B(n)k

⎛⎝

t∫

x

(t − u)g′′(u)du; x

⎞⎠

∥∥∥∥∥∥p

≤ C∥∥∥g

′′∥∥∥pμ2

kp,

where C > 0 is an absolute constant. �

Lemma 3.3 Let A := {A(n)} be a sequence of infinite matrices with nonnegative real entries.Let {Tj } be a sequence of positive linear operators from L p[a, b], 1 ≤ p < ∞, into itself

and assume that (2.2) holds. Then for all k sufficently large and for all g ∈ L(2)p [a, b],

supn

∥∥∥B(n)k (g; x)− g(x)∥∥∥

p≤ C

′p(‖g‖p +

∥∥∥g′′∥∥∥

p)μ2

kp,

where C′p > 0 is independent of f and k.

Proof Let g ∈ L(2)p [a, b] and assume it has been extended outside of [a, b] so that g′′(x) = 0if x /∈ [a, b]. For x, t ∈ [a, b] we know that

g(t)− g(x) = g′(x)(t − x)+

t∫

x

(t − u)g′′(u)du. (3.4)

Using the positivity and linearity of operators Tj and considering (3.4) and Lemma 3.2 weobtain

∥∥∥B(n)k (g; x)− g(x)∥∥∥

p≤

∥∥∥B(n)k (g(t)− g(x); x)∥∥∥

p+ ‖g‖∞

∥∥∥B(n)k (g0; x)− g0(x)∥∥∥

p

≤∥∥∥g

′∥∥∥∞

{ ∥∥∥B(n)k (g1; x)− g1(x)∥∥∥

p

+β∥∥∥B(n)k (g0; x)− g0(x)

∥∥∥p

}

+ C∥∥∥g

′′∥∥∥pμ2

kp + ‖g‖∞∥∥∥B(n)k (g0; x)− g0(x)

∥∥∥p. (3.5)

Applying Theorem 3.1 in [6], from (3.5) we get that

supn

∥∥∥B(n)k (g; x)− g(x)∥∥∥

p≤ C

′p(‖g‖p +

∥∥∥g′′∥∥∥

p)μ2

kp.

This completes the proof. �

123


Theorem 3.4 Let A := {A(n)} be a sequence of infinite matrices with nonnegative realentries. Let {Tj } be a sequence of positive linear operators from L p[a, b], 1 ≤ p < ∞, intoitself, and assume that (2.2) holds. Then for all k sufficently large and for f ∈ L p[a, b] wehave

supn

∥∥∥B(n)k ( f ; x)− f (x)∥∥∥

p≤ C p(min(1, μ2

kp) ‖ f ‖p + w2,p( f ;μkp)),

where C p > 0 is independent of f and k.

Proof Let f ∈ L p[a, b] and g ∈ L(2)p [a, b]. Lemma 3.3 yields

supn

∥∥∥B(n)k ( f ; x)− f (x)∥∥∥

p≤ ‖ f − g‖p sup

n

∑j

a(n)k j

∥∥Tj∥∥

L p→L p

+ supn

∥∥∥B(n)k (g; x)− g(x)∥∥∥

p+ ‖ f − g‖p

≤ (1 + H) ‖ f − g‖p + C′p

(‖g‖p +

∥∥∥g′′∥∥∥

p

)μ2

kp.

Taking infimum over all g ∈ L(2)p [a, b] and using (3.1) and ( 3.2), we have

supn

∥∥∥B(n)k ( f ; x)− f (x)∥∥∥

p≤ C p(min(1, μ2

kp) ‖ f ‖p + w2,p( f ;μkp)),

which concludes the proof. �Corollary 3.5 Let A := {A(n)} be a sequence of infinite matrices with nonnegative realentries. Let {Tj } be a sequence of positive linear operators from L p[a, b], 1 ≤ p < ∞, intoitself. Assume that ( 2.2) holds and μkp → 0, (k → ∞). Then for all f ∈ L p[a, b],we have

limk

supn

∥∥∥B(n)k ( f ; x)− f (x)∥∥∥

p= 0.

Acknowledgments I. Sakaoglu was supported by the Scientific and Technological Research Council ofTurkey (TUBITAK).

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Rate of convergence in $$L_{p}$$ L p approximation