Research ArticleDegree of Approximation of Functions 119891 isin 119867

120596Class by

the (119873119901sdot 119864

1

)Means in the Houmllder Metric

Vishnu Narayan Mishra12 and Kejal Khatri1

1 Applied Mathematics and Humanities Department Sardar Vallabhbhai National Institute of Technology Ichchhanath MahadevDumas Road Surat Gujarat 395 007 India

2 L 1627 Awadh Puri Colony Beniganj Opp ITI Ayodhya Main Road Faizabad Uttar Pradesh 224 001 India

Correspondence should be addressed to Vishnu Narayan Mishra vishnunarayanmishragmailcom

Received 28 December 2013 Revised 28 February 2014 Accepted 28 February 2014 Published 30 April 2014

Academic Editor Ricardo Estrada

Copyright copy 2014 V N Mishra and K Khatri 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

A new estimate for the degree of approximation of a function 119891 isin 119867

120596class by (119873

119901sdot 1198641

) means of its Fourier series has beendetermined Here we extend the results of Singh and Mahajan (2008) which in turn generalize the result of Lal and Yadav (2001)Some corollaries have also been deduced from our main theorem

1 Introduction

The degree of approximation of a function 119891 belongingto various classes using different summability method hasbeen determined by several investigators like Khan [1 2]V N Mishra and L N Mishra [3] Mishra et al [4ndash6] andMishra [7 8] Summability techniques were also applied onsome engineering problems for example Chen and Jeng [9]implemented the Cesaro sum of order (119862 1) and (119862 2) inorder to accelerate the convergence rate to deal with theGibbs phenomenon for the dynamic response of a finiteelastic body subjected to boundary traction Chen and Hong[10] used Cesaro sum regularization technique for hypersingularity of dual integral equation Summability of Fourierseries is useful for engineering analysis for example [11]Recently Mursaleen and Mohiuddine [12] discussed conver-gence methods for double sequences and their applicationsin various fields In sequel Alexits [13] studied the degreeof approximation of the functions in 119867

120572class by the Cesaro

means of their Fourier series in the sup-norm Chandra ([1415]) Mohapatra and Chandra ([16 17]) and Szal [18] havestudied the approximation of functions in Holder metricMishra et al [6] used the technique of approximation of

functions in measuring the errors in the input signals andthe processed output signals In 2008 Singh and Mahajan[19] studied error bound of periodic signals in the Holdermetric and generalized the result of Lal and Yadav [20] undermuch more general assumptions Analysis of signals or timefunctions is of great importance because it conveys infor-mation or attributes of some phenomenon The engineersand scientists use properties of Fourier approximation fordesigning digital filters Especially Psarakis andMoustakides[21] presented a new 119871

2basedmethod for designing the finite

impulse response (FIR) digital filters and got correspondingoptimum approximations having improved performanceWealso discuss an example when the Fourier series of the signalhas Gibbs phenomenon

For a 2120587-periodic signal 119891 isin 119871119901

= 119871119901

[0 2120587] 119901 ge 1periodic integrable in the sense of LebesgueThen the Fourierseries of 119891(119909) is given by

119891 (119909) sim

1198860

2

+

infin

sum

119899=1

(119886119899cos 119899119909 + 119887

119899sin 119899119909)

equiv

infin

sum

119899=0

119860119899(119909)

(1)

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2014 Article ID 837408 9 pageshttpdxdoiorg1011552014837408

2 International Journal of Mathematics and Mathematical Sciences

with (119899 + 1)th partial sum 119904119899(119891 119909) called trigonometric

polynomial of degree (or order) 119899 and given by

119904119899(119891 119909) =

1198860

2

+

119899

sum

119896 =1

(119886119896cos 119896119909 + 119887

119896sin 119896119909)

119899 isin N with 1199040(119891 119909) =

1198860

2

(2)

The conjugate series of Fourier series (1) is given byinfin

sum

119899=1

(119887119899cos 119899119909 minus 119886

119899sin 119899119909) equiv

infin

sum

119899=1

119861119899(119909) (3)

with 119899th partial sum 119904119899(119891 119909)

Let 120596(119905) and 120596lowast(119905) denote two given moduli of continuitysuch that

(120596 (119905))120573120572

= 119874 (120596lowast

(119905)) as 119905 997888rarr 0+

for 0 le 120573 lt 120572 le 1(4)

Let 1198622120587

denote the Banach space of all 2120587-periodic contin-uous functions defined on [minus120587 120587] under the sup-norm Thespace 119871

119901[0 2120587] where 119901 = infin includes the space 119862

2120587 For

some positive constant 119870 the function space 119867120596is defined

by

119867120596= 119891 isin 119862

21205871003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le 119870120596 (

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) (5)

with norm sdot 120596lowast defined by

10038171003817100381710038171198911003817100381710038171003817120596lowast =

10038171003817100381710038171198911003817100381710038171003817119888+ sup119909119910

[Δ120596lowast

119891 (119909 119910)] (6)

where 120596(119905) and 120596lowast(119905) are increasing functions of 11990510038171003817100381710038171198911003817100381710038171003817119888= sup0le119909le2120587

1003816100381610038161003816119891 (119909)

1003816100381610038161003816 (7)

Δ120596lowast

119891 (119909 119910) =

1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

119909 = 119910 (8)

with the understanding that Δ0119891(119909 119910) = 0 If there existpositive constants 119861 and 119870 such that 120596(|119909 minus 119910|) le 119861|119909 minus 119910|

120572

and 120596lowast(|119909 minus 119910|) le 119870 |119909 minus 119910|120573

0 le 120573 lt 120572 le 1 then the space

119867120572= 119891 isin 119862

21205871003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le 119870

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

120572

0 lt 120572 le 1

(9)

is Banach space [22] and themetric induced by the norm sdot 120572

on 119867120572is said to be Holder metric Clearly 119867

120572is a Banach

space which decreases as 120572 increases that is

119867120572sube 119867120573sube 1198622120587 for 0 le 120573 lt 120572 le 1 (10)

Let suminfin119899=0

119886119899be a given infinite series with the sequence of

119899th partial sums 119904119899 Let 119901

119899 be a nonnegative sequence of

constants real or complex and let us write

119875119899=

119899

sum

119896=0

119901119896= 0 forall119899 ge 0

119901minus1= 0 = 119875

minus1 119875119899997888rarr infin as 119899 997888rarr infin

(11)

The sequence to sequence transformation

119905119873

119899(119891 119909) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

119904119896(119891 119909) (12)

defines the sequence 119905119873119899 of Norlund means of the sequence

119904119899 generated by the sequence of coefficients 119901

119899The series

suminfin

119899=0119886119899is said to be summable119873

119901to the sum 119904 if lim

119899rarrinfin119905119873

119899

exists and is equal to 119904In the special case in which

119901119899= (

119899 + 120572 minus 1

120572 minus 1) =

Γ (119899 + 120572)

Γ (119899 + 1) Γ (120572)

(120572 gt 0) (13)

then Norlund summability119873119901reduces to the familiar (119862 120572)

summabilityAn infinite seriessuminfin

119899=0119886119899is said to be (119862 1) summable to

119904 if

(119862 1) =

1

(119899 + 1)

119899

sum

119896=0

119904119896997888rarr 119904 as 119899 997888rarr infin (14)

The 1198641 transform is defined as the 119899th partial sum of 1198641

summability and we denote it by 1198641119899

If

1198641

119899(119891) = 119864

1

119899(119891 119909) =

1

2119899

119899

sum

119896=0

(

119899

119896) 119904119896997888rarr 119904 as 119899 997888rarr infin

(15)

then the infinite series suminfin119899=0

119886119899is said to be (119864 1) summable

to 119904The (119862 1) transform of the (119864 1) transform 119864

1

119899defines

the (119862 1)(119864 1) transform of the partial sums 119904119899of the

series suminfin119899=0

119886119899 that is the product summability (119862 1)(119864 1)

is obtained by superimposing (119862 1) summability on (119864 1)

summability Thus if

(119862119864)1

119899=

1

(119899 + 1)

119899

sum

119896=0

1198641

119896

=

1

(119899 + 1)

119899

sum

119896=0

1

2119896

119896

sum

119903=0

(

119896

119903) 119904119903997888rarr 119904 as 119899 997888rarr infin

(16)

where 1198641119899denotes the (119864 1) transform of 119904

119899 then the series

suminfin

119899=0119886119899with the partial sums 119904

119899is said to be summable

(119862 1)(119864 1) to the definite number 119904 and we can write

(119862119864)1

119899997888rarr 119904 [(119862 1) (119864 1)] as 119899 997888rarr infin (17)

The 119873119901transform of the 1198641 transform defines (119873

119901sdot 1198641

)

product transform and denote it by 119873119864119899(119891) If

119873119864

119899(119891) =

119873119864

119899(119891 119909)

=

1

119875119899

119899

sum

119896=0

119901119899minus119896

1198641

119896(119891) 997888rarr 119904 as 119899 997888rarr infin

(18)

International Journal of Mathematics and Mathematical Sciences 3

then the infinite seriessuminfin119899=0

119886119899is said to be (119873

119901sdot1198641

) summableto 119904

We note that 1198641119899 (119862119864)

1

119899 and

119873119864

119899are also trigonometric

polynomials of degree (or order) 119899The relation between Cesaro mean and Fejer mean is

given by

120590119899(119909) =

1

2120587

int

2120587

0

[119891 (119909 + 119905) + 119891 (119909 minus 119905)] 119865119899(119905) 119889119905 (19)

where the Fejer mean is

1198650(119905) = 1

119865119899(119905) =

1

119899 + 1

(

sin (119899 + 1) (1199052)sin (1199052)

)

2

119905 notin 2120587119885 119899 ge 1

(119899 + 1) 119905 isin 2120587119885

(20)

and Cesaro mean is

120590119899=

1199040(119909) + 119904

1(119909) + sdot sdot sdot + 119904

119899minus1(119909)

119899

=

1

119899

119899minus1

sum

119896=0

119904119896(119909) (21)

Some important particular cases are as follows

(1) (119867 1(119899 + 1)) sdot (1198641) means when 119886

119899119896= 1(119899 minus 119896 +

1) log 119899

(2) (119873 119901119899 119902119899) sdot 1198641means 119886

119899119896= 119901119899minus119896

119902119896119877119899 where 119877

119899=

sum119899

119896=0119901119896119902119899minus119896

= 0

Remark 1 If we take 120596(|119909 minus 119910|) = |119909 minus 119910|120572 then119867

120596reduces

to119867120572class

The conjugate function 119891(119909) is defined for almost every 119909

by

119891 (119909) = minus

1

2120587

int

120587

0

120595 (119905) cot 1199052

119889119905

= limℎrarr0

(minus

1

2120587

int

120587

ℎ

120595 (119905) cot 1199052

119889119905)

(22)

see [23 page 131]We write throughout the paper

120595119909(119905) = 120595 (119905) = 119891 (119909 + 119905) minus 119891 (119909 minus 119905)

120601119909(119905) = 119891 (119909 + 119905) minus 2119891 (119909) + 119891 (119909 minus 119905)

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(23)

We note that the series conjugate to a Fourier series is notnecessarily a Fourier series [23 24] Hence a separate studyof conjugate series is desirable and attracted the attention ofresearchers

2 Known Results

In 2001 Lal and Yadav [20] established the following theoremto estimate the error between the input signal 119891(119909) andthe signal obtained after passing through the (119862 1)(119864 1)-transform

Theorem 2 (see [20]) If a function 119891 119877 rarr 119877 is 2120587-periodicfunction and belongs to class Lip120572 0 lt 120572 le 1 then degreeof approximation by (119862 1)(119864 1) means of its Fourier series isgiven by

100381710038171003817100381710038171003817

1199051198621

sdot1198641

119899(119891) minus 119891 (119909)

100381710038171003817100381710038171003817infin

=

119874(119899minus120572

) 0 lt 120572 lt 1

119874(

log 119899119899

) 120572 = 1

(24)

Recently Singh and Mahajan [19] generalized the above resultunder more general assumptions They proved the following

Theorem 3 (see [19]) Let 120596(119905) defined in (5) be such that

int

120587

119905

120596 (119906)

1199062119889119906 = 119874 (119867 (119905)) 119867 (119905) ge 0

int

119905

0

119867(119906) 119889119906 = 119874 (119905119867 (119905)) as 119905 997888rarr 0+

(25)

then for 0 lt 120573 le 120572 le 1 and 119891 isin 119867120596 we have

100381710038171003817100381710038171003817

1199051198621

sdot1198641

119899(119891) minus 119891 (119909)

100381710038171003817100381710038171003817120596lowast

= 119874(((119899 + 1)minus1

119867(

120587

119899 + 1

))

1minus 120573120572

)

(26)

Theorem4 (see [19]) Consider120596(119905) defined in (5) and for 0 lt120573 le 120572 le 1 and 119891 isin 119867

120596 we have

100381710038171003817100381710038171003817

1199051198621

sdot1198641

119899(119891) minus 119891 (119909)

100381710038171003817100381710038171003817120596lowast

= 119874((120596(

120587

119899

+ 1))

1minus 120573120572

+((119899 + 1)minus1

119899+1

sum

119896=1

120596(

1

119896 + 1

))

1minus 120573120572

)

(27)

3 Main Theorem

In this paper we prove a theorem on the degree of approx-imation of a function

119891(119909) conjugate to a 2120587-periodicfunction 119891 belonging to

119891 isin 119867120596

class by (119873119901sdot 1198641

)

means of conjugate series of its Fourier series This workgeneralizes the results of Singh andMahajan [19] on (119873

119901sdot 1198641

)

summability of conjugate Fourier series We will measure theerror between the input signal 119891(119909) and the processed outputsignal 119873119864

119899(119891 119909) = (1119875

119899) sum119899

119896=0119901119899minus119896

1198641

119896(119891) by establishing the

following theorems

Theorem 5 The functions 120596 satisfy the following conditions

int

120587

119905

120596 (119906)

1199062119889119906 = 119874 (119867 (119905)) 119867 (119905) ge 0 (28)

int

119905

0

119867(119906) 119889119906 = 119874 (119905119867 (119905)) as 119905 997888rarr 0+

(29)

4 International Journal of Mathematics and Mathematical Sciences

where 120596(119905) and 120596lowast(119905) are increasing functions of 119905 Let 119873119901be

the Norlund summability matrix generated by the nonnegative119901119899 such that

(119899 + 1) 119901119899= 119874 (119875

119899) forall119899 ge 0 (30)

Then for 119891 isin 119867120596 0 lt 120573 le 120572 le 1 we have

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(31)

and if 120596(119905) satisfies (28) then for 119891 isin 119867

120596 0 lt 120573 le 120572 le 1 we

have10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(32)

Remark 6 The product transform (119873119901sdot 1198641

) plays an impor-tant role in signal theory as a double digital filter and theoryof Machines in Mechanical Engineering [4 5]

4 Lemmas

In order to prove our main Theorem 5 we require thefollowing lemmas

Lemma 7 If 120595119909(119905) = 120595(119905) = 119891(119909 + 119905) minus 119891(119909 minus 119905) then for

119891 isin 119867

120596 we have

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816le 2119872120596 (

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) (33)

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816le 2119872120596 (|119905|) (34)

It is easy to verify the following

Lemma 8 Let 119901119899 be a nonnegative and nonincreasing

sequence satisfies (30) then for 0 lt 119905 le 120587(119899 + 1) we have

(119873119901sdot 1198641

)119899

(119905) = 119874(

1

119905

) (35)

Proof Consider

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(36)

Using condition (30) sin 119899119905 le 119899119905 sin(1199052) ge (119905120587) for 0 lt

119905 le 120587 and |cos119896(1199052)| le 1 we have

(119873119901sdot 1198641

)119899

(119905)

le

119901119899

119875119899

119899

sum

119896=0

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

= 119874(

1

119905 (119899 + 1)

)[

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)]

= (

1

119905 (119899 + 1)

)

119899

sum

119896=0

Re cos119896 ( 1199052

) 119890119894(119896+1)1199052

= (

1

119905 (119899 + 1)

)Re1198901198941199052 (1 minus cos119899+1 (1199052) 119890 119894(119899+1)1199052

1 minus cos (1199052) 1198901198941199052)

= (

1

1199052(119899 + 1)

)Re 119894 minus 119894 cos (119899 + 1) (1199052) cos119899+1 (1199052)

+ sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

) sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

(119899 + 1) 119905)

= 119874(

1

119905

)

(37)

Lemma 9 Let 119901119899 be a nonnegative and nonincreasing

sequence satisfies (30) then for 120587(119899 + 1) le 119905 le 120587 we get

(119873119901sdot 1198641

)119899

(119905) = 119874(

1

1199052(119899 + 1)

) (38)

Proof Consider

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(39)

Using condition (30) sin (1199052) ge (119905120587) for 0 lt 119905 le 120587| sin 119905| le1 for all 119905 and |cos119896(1199052)| le 1 we have

(119873119901sdot 1198641

)119899

(119905)

le

119901119899

119875119899

119899

sum

119896=0

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

= (

1

119905 (119899 + 1)

)

119899

sum

119896=0

Re cos119896 ( 1199052

) 119890119894(119896+1)1199052

= (

1

119905 (119899 + 1)

)Re1198901198941199052 (1 minus cos119899+1 (1199052) 119890119894(119899+1)1199052

1 minus cos (1199052) 1198901198941199052)

International Journal of Mathematics and Mathematical Sciences 5

= 119874(

1

1199052(119899 + 1)

) sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

)

(40)

Lemma 10 (see [19]) If120596(119905) satisfies conditions (28) and (29)then

int

119906

0

119905minus1

120596 (119905) 119889119905 = 119874 (119906119867 (119906)) as 119906 997888rarr 0+

(41)

5 Proof of Theorem 5

The integral representationof 119904119899(119891 119909) is given by

119904119899(119891 119909) =

infin

sum

119899=1

(119887119899cos 119899119909 minus 119886

119899sin 119899119909)

= minus

1

120587

int

120587

0

120595119909(119905) (

cot (1199052) cos (119899 + 12) 1199052 sin (1199052)

) 119889119905

(42)

and therefore

119904119899(119891 119909) minus

119891 (119909) =

1

2120587

int

120587

0

120595119909(119905)

cos (119899 + 12) 119905sin 1199052

119889119905 (43)

Denoting 1198641 means of 119904119899(119891 119909) by 1198641

119899(119909) we have

1

2119899

119899

sum

119896=0

(

119899

119896) 119904119899(119891 119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

1198641

119899(119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re

119899

sum

119896=0

(

119899

119896) 119890119894(119896+12)119905

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 1198901198941199052(1 + 119890119894119905)

119899

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 2119899cos119899 ( 119905

2

) 119890119894(119899+1)1199052

119889119905

=

1

2120587

int

120587

0

120595119909(119905)

cos119899 (1199052) cos (119899 + 1) (1199052)sin 1199052

119889119905

(44)

Now using condition (30) then 119873119901transform of 1198641 trans-

form is given by

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

=

1

2120587

int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

119901119899minus119896

119875119899

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

)int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

(45)

Set

119864119899(119909) =

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

= 119874(

1

119899 + 1

)

times int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

119864119899(119909 119910)

=1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

= 119874(

1

119899 + 1

)int

120587

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

) [int

120587(119899+1)

0

+int

120587

120587(119899+1)

]

= 1198681+ 1198682 (say)

(46)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198681= 119874 (1) int

120587(119899+1)

0

119905minus1

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(47)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(48)

6 International Journal of Mathematics and Mathematical Sciences

Now using (33) Lemma 8 we get

1198681= 119874(

1

119899 + 1

) int

120587(119899+1)

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

119889119905

= 119874(

1

119899 + 1

) int

120587(119899+1)

0

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

119905

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874 (log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(49)

Now using (33) Lemma 9 we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) 119889119905

= 119874(

1

119899 + 1

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))int

120587

120587(119899+1)

119905minus2

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(50)

Using the fact that we may write 119868119896= 1198681minus120573120572

119896119868120573120572

119896 119896 = 1 2

and combining (47) and (49) we get

1198681= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(51)

Combining (48) and (50) we get

1198682= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(52)

Therefore from (8) (51) and (52) we have

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= sup119909119910

1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(53)

Since1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= sup0le119909le2120587

100381610038161003816100381610038161003816

119879sdot1198641

119899(119891) minus

119891 (119909)

100381610038161003816100381610038161003816

(54)

it follows from (47) and (48) that

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874((119899 + 1)

minus1

119867(

120587

119899 + 1

)) (55)

Combining (53) and (55) we get

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times[(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(56)

To prove (32) if 120596(119905) satisfies only (28) then using (34) andsecond mean value theorem for integrals we have

1198681= 119874 (120596 (120587 (119899 + 1))) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

(57)

Combining (49) and (57) we get

1198681= 119874([log (119899 + 1) 120596 ( 120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(58)

Again if 120596(119905) satisfies only (28) then using (34) and secondmean value theorem for integrals we have

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

120596 (119905)

1199052119889119905

= 119874(

1

119899 + 1

)int

119899+1

1

120596(

120587

119905

) 119889119905

= 119874(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)

(59)

Combining (50) and (59) we get

1198682= 119874([(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(60)

International Journal of Mathematics and Mathematical Sciences 7

Therefore

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times ( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(61)

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

+ 119874((

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

))

(62)

Combining (61) and (62) we get10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times (log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(63)

This completes the proof of Theorem 5

6 Applications

The theory of approximation is a very extensive field andthe study of theory of trigonometric approximation is ofgreatmathematical interest and of great practical importanceAs mentioned in [21] the 119871

119901space in general and 119871

2and

119871infin

in particular play an important role in the theory ofsignals and filters From the point of view of the applicationssharper estimates of infinite matrices [25] are useful to getbounds for the lattice norms (which occur in solid statephysics) of matrix valued functions and enable to investigateperturbations of matrix valued functions and compare themThe following corollaries may be derived fromTheorem 5

If 120596(|119909 minus 119910|) le 119861|119909 minus 119910|120572

120596lowast

(|119909 minus 119910|) le 119870|119909 minus 119910|120573

0 le

120573 lt 120572 le 1 and set

119867(119906) =

119906120572minus1

0 lt 120572 lt 1

log(1119906

) 120572 = 1

(64)

then we get Corollary 11

Corollary 11 If 119891 isin 119867120572 0 lt 120573 le 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120573

=

119874((log (119899 + 1))120573120572(119899 + 1)120573minus120572) 0 lt 120572 lt 1

119874 (log (119899 + 1) (119899 + 1)120573minus1) 120572 = 1

(65)

If we put 120573 = 0 in Corollary 11 then we get Corollary 12

Corollary 12 If 119891 isin Lip120572 0 lt 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817119888=

119874((119899 + 1)minus120572

) 0 lt 120572 lt 1

119874(

log (119899 + 1)(119899 + 1)

) 120572 = 1

(66)

7 Example

In the example we see how the sequence of averages (ie1205901

119899(119909)-means or (119862 1)mean) andNorlundmean119873

119901of partial

sums of a Fourier series is better behaved than the sequenceof partial sums 119904

119899(119909) itself

Let

119891 (119909) =

minus1 minus120587 le 119909 lt 0

1 0 le 119909 lt 120587

(67)

with 119891(119909 + 2120587) = 119891(119909) for all real 119909 Fourier series of 119891(119909) isgiven by

2

120587

infin

sum

119899=1

1 minus (minus1)119899

119899

sin 119899119909 minus120587 le 119909 le 120587 (68)

Then 119899th partial sum 119904119899(119909) of Fourier series (68) and 119899th

Cesaro sum for 120575 = 1 that is 1205901119899(119909) for the series (68) are

given by

119904119899(119909) =

4

120587

(sin119909 + 1

3

sin 3119909 + sdot sdot sdot + 1

119899

sin 119899119909)

1205901

119899(119909) =

2

120587

119899

sum

119896=1

(1 minus

119896

119899

)(

1 minus (minus1)119896

119896

) sin 119896119909(69)

FromTheorem 20 of Hardyrsquos ldquoDivergent Seriesrdquo if a Norlundmethod 119873

119901has increasing weights 119901

119899 then it is stronger

than (119862 1)Now take119873

119901to be the Norlundmatrix generated by 119901

119899=

119899 + 1 then Norlund means119873119901is given by

119905119873

119899(119891 119909) =

2

(119899 + 1) (119899 + 2)

119899

sum

119896=0

(119899 minus 119896 + 1) 119904119896(119891 119909) (70)

In this graph we observe that 1205901119899(119909) 119905119873

119899(119891 119909) converges to

119891(119909) faster than 119904119899(119909) in the interval [minus120587 120587] We further note

that near the points of discontinuities that is minus120587 0 and 120587the graph of 119904

5and 11990410

shows peaks that move closer to theline passing through points of discontinuity as n increases(Gibbs phenomenon) but in the graph of 1205901

119899(119909) 119905119873

119899(119891 119909)

8 International Journal of Mathematics and Mathematical Sciences

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(a)

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(b)

Figure 1 Graph of 119891(119909) (blue) 119904119899(119909) (pink) 1205901

119899(119909) (yellow) 119905119873

119899(119891 119909) (green) 119899 = 5 and 10

119899 = 5 10 the peaks becomeflatter (Figure 1)TheGibbs phe-nomenon is an overshoot a peculiarity of the Fourier seriesand other eigen function series at a simple discontinuitythat is the convergence of Fourier series is very slow nearthe point of discontinuity Thus the product summabilitymeans of the Fourier series of 119891(119909) overshoot the GibbsPhenomenon and show the smoothing effect of the methodThus 1205901

119899(119909) 119905119873

119899(119891 119909) is the better approximant than 119904

119899(119909) and

119873119901method is stronger than (119862 1) method

8 Conclusion

Several results concerning the degree of approximation ofperiodic signals (functions) by product summability meansof Fourier series and conjugate Fourier series in generalizedHolder metric and Holder metric have been reviewed Usinggraphical representation 120590

1

119899(119909) 119905119873

119899(119891 119909) is a better approxi-

mant to 119904119899(119909) but till now nothing has been done to show

this Some interesting application of the operator (119873119901sdot 1198641

)

used in this paper is pointed out in Remark 6 Also the resultof our theorem is more general rather than the results of anyother previous proved theorems whichwill enrich the literateof summability theory of infinite series

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude tothe anonymous learned referee(s) and the editor for theirvaluable suggestions and constructive comments whichresulted in the subsequent improvement of this paper Specialthanks are due to Professor Ricardo Estrada Academic Editorof IJMMS and Professor Mohammed Mahmoud Editorial

office Hindawi Publishing Corporation for kind cooper-ation smooth behavior during communication and theirefforts to send the reports of the paper timelyThe authors arealso grateful to all the editorial boardmembers and reviewersof Hindawi prestigious journal that is International Journalof Mathematics and Mathematical Sciences The secondauthor Kejal Khatri acknowledges the Ministry of HumanResource Development New Delhi India for providing thefinancial support and to carry out her research work underthe guidance of Dr Vishnu Narayan Mishra at SVNITSurat (Gujarat) India The first author Vishnu NarayanMishra acknowledges Cumulative Professional DevelopmentAllowance (CPDA) SVNIT Surat (Gujarat) India for sup-porting this research paper Both authors carried out theproof All the authors conceived of the study and participatedin its design and coordination Both the authors read andapproved the final manuscript

References

[1] H H Khan ldquoOn the degree of approximation of a functionsbelonging to the class Lip(120572 p)rdquo Indian Journal of Pure andApplied Mathematics vol 5 pp 132ndash136 1974

[2] H H Khan Approximation of classes of functions [PhD thesis]AMU Aligarh India 1974

[3] V N Mishra and L N Mishra ldquoTrigonometric approximationin 119871119901(119901 ge 1) spacesrdquo International Journal of Contemporary

Mathematical Sciences vol 7 pp 909ndash918 2012[4] V N Mishra K Khatri and L N Mishra ldquoProduct

(119873 119901119899)(119862 1) summability of a sequence of Fourier coefficientsrdquo

Mathematical Sciences vol 6 p 38 2012[5] VNMishra KKhatri and LNMishra ldquoProduct summability

transform of conjugate series of Fourier seriesrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2012Article ID 298923 13 pages 2012

[6] V N Mishra H H Khan K Khatri and L N Mishra ldquoOnapproximation of conjugate of signals (Functions) belongingto the generalized weighted 119882(119871

119903 120585(119905)) (119903 ge 1)-class by

product summability means of conjugate series of fourier

International Journal of Mathematics and Mathematical Sciences 9

seriesrdquo International Journal of Mathematical Analysis vol 6no 35 pp 1703ndash1715 2012

[7] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to the weighted 119882(119871

119901 120585(119905)) (119901 ge 1)-

class by almost matrix summability method of its conjugateFourier seriesrdquo International Journal of Applied Mathematicsand Mechanics vol 5 no 7 pp 16ndash27 2009

[8] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to Generalized Weighted 119882(119871

119901 120585(119905))

(119901 ge 1)-class by product summability methodrdquo Journal ofInternational Academy of Physical Sciences vol 14 no 4 pp413ndash423 2010

[9] J T Chen and Y S Jeng ldquoDual series representation and itsapplications to a string subjected to support motionsrdquoAdvancesin Engineering Software vol 27 no 3 pp 227ndash238 1996

[10] J T Chen and H-K Hong ldquoReview of dual boundary elementmethods with emphasis on hypersingular integrals and diver-gent seriesrdquo Applied Mechanics Reviews vol 52 no 1 pp 17ndash321999

[11] J T Chen H-K Hong C S Yeh and S W Chyuan ldquoInte-gral representations and regularizations for a divergent seriessolution of a beam subjected to support motionsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 9 pp 909ndash925 1996

[12] M Mursaleen and S A Mohiuddine Convergence Methods ForDouble Sequences and Applications Springer 2014

[13] G Alexits Convergence Problems of Orthogonal Series Perga-mon Elmsford New York NY USA 1961

[14] P Chandra ldquoOn the degree of approximation of a class of func-tions by means of Fourier seriesrdquo Acta Mathematica Hungaricavol 52 no 3-4 pp 199ndash205 1988

[15] P Chandra ldquoA note on the degree of approximation of contin-uous functionsrdquo Acta Mathematica Hungarica vol 62 no 1-2pp 21ndash23 1993

[16] R NMohapatra and P Chandra ldquoHolder continuous functionsand their Euler Borel and Taylor meansrdquo Math Chronicle vol11 pp 81ndash96 1982

[17] R N Mohapatra and P Chandra ldquoDegree of approximation offunctions in the Hlder metricrdquo Acta Mathematica Hungaricavol 41 no 1-2 pp 67ndash76 1983

[18] B Szal ldquoOn the rate of strong summability by matrix means inthe generalized Holder metricrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 1 article 28 2008

[19] T Singh and P Mahajan ldquoError bound of periodic signals inthe Holder metricrdquo International Journal of Mathematics andMathematical Science vol 2008 Article ID 495075 9 pages2008

[20] S Lal and K N S Yadav ldquoOn degree of approximation offunction belonging to the Lipschitz class by (119862 1)(119864 1)meansof its Fourier seriesrdquo Bulletin of Calcutta Mathematical Societyvol 93 pp 191ndash196 2001

[21] E Z Psarakis and G V Moustakides ldquoAn L2-based method

for the design of 1-D zero phase FIR digital filtersrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 44 no 7 pp 591ndash601 1997

[22] G Das T Ghosh and B K Ray ldquoDegree of approximation offunctions in the Holder metric by (e c) Meansrdquo Proceedings ofthe Indian Academy of Science vol 105 pp 315ndash327 1995

[23] A ZygmundTrigonometric Series CambridgeUniversity PressLondon UK 3rd edition 2002

[24] G Bachman L Narici and E Beckenstien Fourier andWaveletAnalysis Springer New York NY USA 2000

[25] M I Gilrsquo ldquoEstimates for entries of matrix valued functions ofinfinite matricesrdquoMathematical Physics Analysis and Geometryvol 11 no 2 pp 175ndash186 2008

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2 International Journal of Mathematics and Mathematical Sciences

with (119899 + 1)th partial sum 119904119899(119891 119909) called trigonometric

polynomial of degree (or order) 119899 and given by

119904119899(119891 119909) =

1198860

2

+

119899

sum

119896 =1

(119886119896cos 119896119909 + 119887

119896sin 119896119909)

119899 isin N with 1199040(119891 119909) =

1198860

2

(2)

The conjugate series of Fourier series (1) is given byinfin

sum

119899=1

(119887119899cos 119899119909 minus 119886

119899sin 119899119909) equiv

infin

sum

119899=1

119861119899(119909) (3)

with 119899th partial sum 119904119899(119891 119909)

Let 120596(119905) and 120596lowast(119905) denote two given moduli of continuitysuch that

(120596 (119905))120573120572

= 119874 (120596lowast

(119905)) as 119905 997888rarr 0+

for 0 le 120573 lt 120572 le 1(4)

Let 1198622120587

denote the Banach space of all 2120587-periodic contin-uous functions defined on [minus120587 120587] under the sup-norm Thespace 119871

119901[0 2120587] where 119901 = infin includes the space 119862

2120587 For

some positive constant 119870 the function space 119867120596is defined

by

119867120596= 119891 isin 119862

21205871003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le 119870120596 (

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) (5)

with norm sdot 120596lowast defined by

10038171003817100381710038171198911003817100381710038171003817120596lowast =

10038171003817100381710038171198911003817100381710038171003817119888+ sup119909119910

[Δ120596lowast

119891 (119909 119910)] (6)

where 120596(119905) and 120596lowast(119905) are increasing functions of 11990510038171003817100381710038171198911003817100381710038171003817119888= sup0le119909le2120587

1003816100381610038161003816119891 (119909)

1003816100381610038161003816 (7)

Δ120596lowast

119891 (119909 119910) =

1003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

119909 = 119910 (8)

with the understanding that Δ0119891(119909 119910) = 0 If there existpositive constants 119861 and 119870 such that 120596(|119909 minus 119910|) le 119861|119909 minus 119910|

120572

and 120596lowast(|119909 minus 119910|) le 119870 |119909 minus 119910|120573

0 le 120573 lt 120572 le 1 then the space

119867120572= 119891 isin 119862

21205871003816100381610038161003816119891 (119909) minus 119891 (119910)

1003816100381610038161003816le 119870

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816

120572

0 lt 120572 le 1

(9)

is Banach space [22] and themetric induced by the norm sdot 120572

on 119867120572is said to be Holder metric Clearly 119867

120572is a Banach

space which decreases as 120572 increases that is

119867120572sube 119867120573sube 1198622120587 for 0 le 120573 lt 120572 le 1 (10)

Let suminfin119899=0

119886119899be a given infinite series with the sequence of

119899th partial sums 119904119899 Let 119901

119899 be a nonnegative sequence of

constants real or complex and let us write

119875119899=

119899

sum

119896=0

119901119896= 0 forall119899 ge 0

119901minus1= 0 = 119875

minus1 119875119899997888rarr infin as 119899 997888rarr infin

(11)

The sequence to sequence transformation

119905119873

119899(119891 119909) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

119904119896(119891 119909) (12)

defines the sequence 119905119873119899 of Norlund means of the sequence

119904119899 generated by the sequence of coefficients 119901

119899The series

suminfin

119899=0119886119899is said to be summable119873

119901to the sum 119904 if lim

119899rarrinfin119905119873

119899

exists and is equal to 119904In the special case in which

119901119899= (

119899 + 120572 minus 1

120572 minus 1) =

Γ (119899 + 120572)

Γ (119899 + 1) Γ (120572)

(120572 gt 0) (13)

then Norlund summability119873119901reduces to the familiar (119862 120572)

summabilityAn infinite seriessuminfin

119899=0119886119899is said to be (119862 1) summable to

119904 if

(119862 1) =

1

(119899 + 1)

119899

sum

119896=0

119904119896997888rarr 119904 as 119899 997888rarr infin (14)

The 1198641 transform is defined as the 119899th partial sum of 1198641

summability and we denote it by 1198641119899

If

1198641

119899(119891) = 119864

1

119899(119891 119909) =

1

2119899

119899

sum

119896=0

(

119899

119896) 119904119896997888rarr 119904 as 119899 997888rarr infin

(15)

then the infinite series suminfin119899=0

119886119899is said to be (119864 1) summable

to 119904The (119862 1) transform of the (119864 1) transform 119864

1

119899defines

the (119862 1)(119864 1) transform of the partial sums 119904119899of the

series suminfin119899=0

119886119899 that is the product summability (119862 1)(119864 1)

is obtained by superimposing (119862 1) summability on (119864 1)

summability Thus if

(119862119864)1

119899=

1

(119899 + 1)

119899

sum

119896=0

1198641

119896

=

1

(119899 + 1)

119899

sum

119896=0

1

2119896

119896

sum

119903=0

(

119896

119903) 119904119903997888rarr 119904 as 119899 997888rarr infin

(16)

where 1198641119899denotes the (119864 1) transform of 119904

119899 then the series

suminfin

119899=0119886119899with the partial sums 119904

119899is said to be summable

(119862 1)(119864 1) to the definite number 119904 and we can write

(119862119864)1

119899997888rarr 119904 [(119862 1) (119864 1)] as 119899 997888rarr infin (17)

The 119873119901transform of the 1198641 transform defines (119873

119901sdot 1198641

)

product transform and denote it by 119873119864119899(119891) If

119873119864

119899(119891) =

119873119864

119899(119891 119909)

=

1

119875119899

119899

sum

119896=0

119901119899minus119896

1198641

119896(119891) 997888rarr 119904 as 119899 997888rarr infin

(18)

International Journal of Mathematics and Mathematical Sciences 3

then the infinite seriessuminfin119899=0

119886119899is said to be (119873

119901sdot1198641

) summableto 119904

We note that 1198641119899 (119862119864)

1

119899 and

119873119864

119899are also trigonometric

polynomials of degree (or order) 119899The relation between Cesaro mean and Fejer mean is

given by

120590119899(119909) =

1

2120587

int

2120587

0

[119891 (119909 + 119905) + 119891 (119909 minus 119905)] 119865119899(119905) 119889119905 (19)

where the Fejer mean is

1198650(119905) = 1

119865119899(119905) =

1

119899 + 1

(

sin (119899 + 1) (1199052)sin (1199052)

)

2

119905 notin 2120587119885 119899 ge 1

(119899 + 1) 119905 isin 2120587119885

(20)

and Cesaro mean is

120590119899=

1199040(119909) + 119904

1(119909) + sdot sdot sdot + 119904

119899minus1(119909)

119899

=

1

119899

119899minus1

sum

119896=0

119904119896(119909) (21)

Some important particular cases are as follows

(1) (119867 1(119899 + 1)) sdot (1198641) means when 119886

119899119896= 1(119899 minus 119896 +

1) log 119899

(2) (119873 119901119899 119902119899) sdot 1198641means 119886

119899119896= 119901119899minus119896

119902119896119877119899 where 119877

119899=

sum119899

119896=0119901119896119902119899minus119896

= 0

Remark 1 If we take 120596(|119909 minus 119910|) = |119909 minus 119910|120572 then119867

120596reduces

to119867120572class

The conjugate function 119891(119909) is defined for almost every 119909

by

119891 (119909) = minus

1

2120587

int

120587

0

120595 (119905) cot 1199052

119889119905

= limℎrarr0

(minus

1

2120587

int

120587

ℎ

120595 (119905) cot 1199052

119889119905)

(22)

see [23 page 131]We write throughout the paper

120595119909(119905) = 120595 (119905) = 119891 (119909 + 119905) minus 119891 (119909 minus 119905)

120601119909(119905) = 119891 (119909 + 119905) minus 2119891 (119909) + 119891 (119909 minus 119905)

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(23)

We note that the series conjugate to a Fourier series is notnecessarily a Fourier series [23 24] Hence a separate studyof conjugate series is desirable and attracted the attention ofresearchers

2 Known Results

In 2001 Lal and Yadav [20] established the following theoremto estimate the error between the input signal 119891(119909) andthe signal obtained after passing through the (119862 1)(119864 1)-transform

Theorem 2 (see [20]) If a function 119891 119877 rarr 119877 is 2120587-periodicfunction and belongs to class Lip120572 0 lt 120572 le 1 then degreeof approximation by (119862 1)(119864 1) means of its Fourier series isgiven by

100381710038171003817100381710038171003817

1199051198621

sdot1198641

119899(119891) minus 119891 (119909)

100381710038171003817100381710038171003817infin

=

119874(119899minus120572

) 0 lt 120572 lt 1

119874(

log 119899119899

) 120572 = 1

(24)

Recently Singh and Mahajan [19] generalized the above resultunder more general assumptions They proved the following

Theorem 3 (see [19]) Let 120596(119905) defined in (5) be such that

int

120587

119905

120596 (119906)

1199062119889119906 = 119874 (119867 (119905)) 119867 (119905) ge 0

int

119905

0

119867(119906) 119889119906 = 119874 (119905119867 (119905)) as 119905 997888rarr 0+

(25)

then for 0 lt 120573 le 120572 le 1 and 119891 isin 119867120596 we have

100381710038171003817100381710038171003817

1199051198621

sdot1198641

119899(119891) minus 119891 (119909)

100381710038171003817100381710038171003817120596lowast

= 119874(((119899 + 1)minus1

119867(

120587

119899 + 1

))

1minus 120573120572

)

(26)

Theorem4 (see [19]) Consider120596(119905) defined in (5) and for 0 lt120573 le 120572 le 1 and 119891 isin 119867

120596 we have

100381710038171003817100381710038171003817

1199051198621

sdot1198641

119899(119891) minus 119891 (119909)

100381710038171003817100381710038171003817120596lowast

= 119874((120596(

120587

119899

+ 1))

1minus 120573120572

+((119899 + 1)minus1

119899+1

sum

119896=1

120596(

1

119896 + 1

))

1minus 120573120572

)

(27)

3 Main Theorem

In this paper we prove a theorem on the degree of approx-imation of a function

119891(119909) conjugate to a 2120587-periodicfunction 119891 belonging to

119891 isin 119867120596

class by (119873119901sdot 1198641

)

means of conjugate series of its Fourier series This workgeneralizes the results of Singh andMahajan [19] on (119873

119901sdot 1198641

)

summability of conjugate Fourier series We will measure theerror between the input signal 119891(119909) and the processed outputsignal 119873119864

119899(119891 119909) = (1119875

119899) sum119899

119896=0119901119899minus119896

1198641

119896(119891) by establishing the

following theorems

Theorem 5 The functions 120596 satisfy the following conditions

int

120587

119905

120596 (119906)

1199062119889119906 = 119874 (119867 (119905)) 119867 (119905) ge 0 (28)

int

119905

0

119867(119906) 119889119906 = 119874 (119905119867 (119905)) as 119905 997888rarr 0+

(29)

4 International Journal of Mathematics and Mathematical Sciences

where 120596(119905) and 120596lowast(119905) are increasing functions of 119905 Let 119873119901be

the Norlund summability matrix generated by the nonnegative119901119899 such that

(119899 + 1) 119901119899= 119874 (119875

119899) forall119899 ge 0 (30)

Then for 119891 isin 119867120596 0 lt 120573 le 120572 le 1 we have

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(31)

and if 120596(119905) satisfies (28) then for 119891 isin 119867

120596 0 lt 120573 le 120572 le 1 we

have10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(32)

Remark 6 The product transform (119873119901sdot 1198641

) plays an impor-tant role in signal theory as a double digital filter and theoryof Machines in Mechanical Engineering [4 5]

4 Lemmas

In order to prove our main Theorem 5 we require thefollowing lemmas

Lemma 7 If 120595119909(119905) = 120595(119905) = 119891(119909 + 119905) minus 119891(119909 minus 119905) then for

119891 isin 119867

120596 we have

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816le 2119872120596 (

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) (33)

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816le 2119872120596 (|119905|) (34)

It is easy to verify the following

Lemma 8 Let 119901119899 be a nonnegative and nonincreasing

sequence satisfies (30) then for 0 lt 119905 le 120587(119899 + 1) we have

(119873119901sdot 1198641

)119899

(119905) = 119874(

1

119905

) (35)

Proof Consider

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(36)

Using condition (30) sin 119899119905 le 119899119905 sin(1199052) ge (119905120587) for 0 lt

119905 le 120587 and |cos119896(1199052)| le 1 we have

(119873119901sdot 1198641

)119899

(119905)

le

119901119899

119875119899

119899

sum

119896=0

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

= 119874(

1

119905 (119899 + 1)

)[

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)]

= (

1

119905 (119899 + 1)

)

119899

sum

119896=0

Re cos119896 ( 1199052

) 119890119894(119896+1)1199052

= (

1

119905 (119899 + 1)

)Re1198901198941199052 (1 minus cos119899+1 (1199052) 119890 119894(119899+1)1199052

1 minus cos (1199052) 1198901198941199052)

= (

1

1199052(119899 + 1)

)Re 119894 minus 119894 cos (119899 + 1) (1199052) cos119899+1 (1199052)

+ sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

) sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

(119899 + 1) 119905)

= 119874(

1

119905

)

(37)

Lemma 9 Let 119901119899 be a nonnegative and nonincreasing

sequence satisfies (30) then for 120587(119899 + 1) le 119905 le 120587 we get

(119873119901sdot 1198641

)119899

(119905) = 119874(

1

1199052(119899 + 1)

) (38)

Proof Consider

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(39)

Using condition (30) sin (1199052) ge (119905120587) for 0 lt 119905 le 120587| sin 119905| le1 for all 119905 and |cos119896(1199052)| le 1 we have

(119873119901sdot 1198641

)119899

(119905)

le

119901119899

119875119899

119899

sum

119896=0

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

= (

1

119905 (119899 + 1)

)

119899

sum

119896=0

Re cos119896 ( 1199052

) 119890119894(119896+1)1199052

= (

1

119905 (119899 + 1)

)Re1198901198941199052 (1 minus cos119899+1 (1199052) 119890119894(119899+1)1199052

1 minus cos (1199052) 1198901198941199052)

International Journal of Mathematics and Mathematical Sciences 5

= 119874(

1

1199052(119899 + 1)

) sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

)

(40)

Lemma 10 (see [19]) If120596(119905) satisfies conditions (28) and (29)then

int

119906

0

119905minus1

120596 (119905) 119889119905 = 119874 (119906119867 (119906)) as 119906 997888rarr 0+

(41)

5 Proof of Theorem 5

The integral representationof 119904119899(119891 119909) is given by

119904119899(119891 119909) =

infin

sum

119899=1

(119887119899cos 119899119909 minus 119886

119899sin 119899119909)

= minus

1

120587

int

120587

0

120595119909(119905) (

cot (1199052) cos (119899 + 12) 1199052 sin (1199052)

) 119889119905

(42)

and therefore

119904119899(119891 119909) minus

119891 (119909) =

1

2120587

int

120587

0

120595119909(119905)

cos (119899 + 12) 119905sin 1199052

119889119905 (43)

Denoting 1198641 means of 119904119899(119891 119909) by 1198641

119899(119909) we have

1

2119899

119899

sum

119896=0

(

119899

119896) 119904119899(119891 119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

1198641

119899(119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re

119899

sum

119896=0

(

119899

119896) 119890119894(119896+12)119905

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 1198901198941199052(1 + 119890119894119905)

119899

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 2119899cos119899 ( 119905

2

) 119890119894(119899+1)1199052

119889119905

=

1

2120587

int

120587

0

120595119909(119905)

cos119899 (1199052) cos (119899 + 1) (1199052)sin 1199052

119889119905

(44)

Now using condition (30) then 119873119901transform of 1198641 trans-

form is given by

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

=

1

2120587

int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

119901119899minus119896

119875119899

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

)int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

(45)

Set

119864119899(119909) =

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

= 119874(

1

119899 + 1

)

times int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

119864119899(119909 119910)

=1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

= 119874(

1

119899 + 1

)int

120587

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

) [int

120587(119899+1)

0

+int

120587

120587(119899+1)

]

= 1198681+ 1198682 (say)

(46)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198681= 119874 (1) int

120587(119899+1)

0

119905minus1

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(47)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(48)

6 International Journal of Mathematics and Mathematical Sciences

Now using (33) Lemma 8 we get

1198681= 119874(

1

119899 + 1

) int

120587(119899+1)

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

119889119905

= 119874(

1

119899 + 1

) int

120587(119899+1)

0

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

119905

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874 (log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(49)

Now using (33) Lemma 9 we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) 119889119905

= 119874(

1

119899 + 1

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))int

120587

120587(119899+1)

119905minus2

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(50)

Using the fact that we may write 119868119896= 1198681minus120573120572

119896119868120573120572

119896 119896 = 1 2

and combining (47) and (49) we get

1198681= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(51)

Combining (48) and (50) we get

1198682= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(52)

Therefore from (8) (51) and (52) we have

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= sup119909119910

1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(53)

Since1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= sup0le119909le2120587

100381610038161003816100381610038161003816

119879sdot1198641

119899(119891) minus

119891 (119909)

100381610038161003816100381610038161003816

(54)

it follows from (47) and (48) that

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874((119899 + 1)

minus1

119867(

120587

119899 + 1

)) (55)

Combining (53) and (55) we get

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times[(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(56)

To prove (32) if 120596(119905) satisfies only (28) then using (34) andsecond mean value theorem for integrals we have

1198681= 119874 (120596 (120587 (119899 + 1))) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

(57)

Combining (49) and (57) we get

1198681= 119874([log (119899 + 1) 120596 ( 120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(58)

Again if 120596(119905) satisfies only (28) then using (34) and secondmean value theorem for integrals we have

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

120596 (119905)

1199052119889119905

= 119874(

1

119899 + 1

)int

119899+1

1

120596(

120587

119905

) 119889119905

= 119874(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)

(59)

Combining (50) and (59) we get

1198682= 119874([(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(60)

International Journal of Mathematics and Mathematical Sciences 7

Therefore

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times ( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(61)

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

+ 119874((

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

))

(62)

Combining (61) and (62) we get10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times (log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(63)

This completes the proof of Theorem 5

6 Applications

The theory of approximation is a very extensive field andthe study of theory of trigonometric approximation is ofgreatmathematical interest and of great practical importanceAs mentioned in [21] the 119871

119901space in general and 119871

2and

119871infin

in particular play an important role in the theory ofsignals and filters From the point of view of the applicationssharper estimates of infinite matrices [25] are useful to getbounds for the lattice norms (which occur in solid statephysics) of matrix valued functions and enable to investigateperturbations of matrix valued functions and compare themThe following corollaries may be derived fromTheorem 5

If 120596(|119909 minus 119910|) le 119861|119909 minus 119910|120572

120596lowast

(|119909 minus 119910|) le 119870|119909 minus 119910|120573

0 le

120573 lt 120572 le 1 and set

119867(119906) =

119906120572minus1

0 lt 120572 lt 1

log(1119906

) 120572 = 1

(64)

then we get Corollary 11

Corollary 11 If 119891 isin 119867120572 0 lt 120573 le 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120573

=

119874((log (119899 + 1))120573120572(119899 + 1)120573minus120572) 0 lt 120572 lt 1

119874 (log (119899 + 1) (119899 + 1)120573minus1) 120572 = 1

(65)

If we put 120573 = 0 in Corollary 11 then we get Corollary 12

Corollary 12 If 119891 isin Lip120572 0 lt 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817119888=

119874((119899 + 1)minus120572

) 0 lt 120572 lt 1

119874(

log (119899 + 1)(119899 + 1)

) 120572 = 1

(66)

7 Example

In the example we see how the sequence of averages (ie1205901

119899(119909)-means or (119862 1)mean) andNorlundmean119873

119901of partial

sums of a Fourier series is better behaved than the sequenceof partial sums 119904

119899(119909) itself

Let

119891 (119909) =

minus1 minus120587 le 119909 lt 0

1 0 le 119909 lt 120587

(67)

with 119891(119909 + 2120587) = 119891(119909) for all real 119909 Fourier series of 119891(119909) isgiven by

2

120587

infin

sum

119899=1

1 minus (minus1)119899

119899

sin 119899119909 minus120587 le 119909 le 120587 (68)

Then 119899th partial sum 119904119899(119909) of Fourier series (68) and 119899th

Cesaro sum for 120575 = 1 that is 1205901119899(119909) for the series (68) are

given by

119904119899(119909) =

4

120587

(sin119909 + 1

3

sin 3119909 + sdot sdot sdot + 1

119899

sin 119899119909)

1205901

119899(119909) =

2

120587

119899

sum

119896=1

(1 minus

119896

119899

)(

1 minus (minus1)119896

119896

) sin 119896119909(69)

FromTheorem 20 of Hardyrsquos ldquoDivergent Seriesrdquo if a Norlundmethod 119873

119901has increasing weights 119901

119899 then it is stronger

than (119862 1)Now take119873

119901to be the Norlundmatrix generated by 119901

119899=

119899 + 1 then Norlund means119873119901is given by

119905119873

119899(119891 119909) =

2

(119899 + 1) (119899 + 2)

119899

sum

119896=0

(119899 minus 119896 + 1) 119904119896(119891 119909) (70)

In this graph we observe that 1205901119899(119909) 119905119873

119899(119891 119909) converges to

119891(119909) faster than 119904119899(119909) in the interval [minus120587 120587] We further note

that near the points of discontinuities that is minus120587 0 and 120587the graph of 119904

5and 11990410

shows peaks that move closer to theline passing through points of discontinuity as n increases(Gibbs phenomenon) but in the graph of 1205901

119899(119909) 119905119873

119899(119891 119909)

8 International Journal of Mathematics and Mathematical Sciences

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(a)

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(b)

Figure 1 Graph of 119891(119909) (blue) 119904119899(119909) (pink) 1205901

119899(119909) (yellow) 119905119873

119899(119891 119909) (green) 119899 = 5 and 10

119899 = 5 10 the peaks becomeflatter (Figure 1)TheGibbs phe-nomenon is an overshoot a peculiarity of the Fourier seriesand other eigen function series at a simple discontinuitythat is the convergence of Fourier series is very slow nearthe point of discontinuity Thus the product summabilitymeans of the Fourier series of 119891(119909) overshoot the GibbsPhenomenon and show the smoothing effect of the methodThus 1205901

119899(119909) 119905119873

119899(119891 119909) is the better approximant than 119904

119899(119909) and

119873119901method is stronger than (119862 1) method

8 Conclusion

Several results concerning the degree of approximation ofperiodic signals (functions) by product summability meansof Fourier series and conjugate Fourier series in generalizedHolder metric and Holder metric have been reviewed Usinggraphical representation 120590

1

119899(119909) 119905119873

119899(119891 119909) is a better approxi-

mant to 119904119899(119909) but till now nothing has been done to show

this Some interesting application of the operator (119873119901sdot 1198641

)

used in this paper is pointed out in Remark 6 Also the resultof our theorem is more general rather than the results of anyother previous proved theorems whichwill enrich the literateof summability theory of infinite series

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude tothe anonymous learned referee(s) and the editor for theirvaluable suggestions and constructive comments whichresulted in the subsequent improvement of this paper Specialthanks are due to Professor Ricardo Estrada Academic Editorof IJMMS and Professor Mohammed Mahmoud Editorial

office Hindawi Publishing Corporation for kind cooper-ation smooth behavior during communication and theirefforts to send the reports of the paper timelyThe authors arealso grateful to all the editorial boardmembers and reviewersof Hindawi prestigious journal that is International Journalof Mathematics and Mathematical Sciences The secondauthor Kejal Khatri acknowledges the Ministry of HumanResource Development New Delhi India for providing thefinancial support and to carry out her research work underthe guidance of Dr Vishnu Narayan Mishra at SVNITSurat (Gujarat) India The first author Vishnu NarayanMishra acknowledges Cumulative Professional DevelopmentAllowance (CPDA) SVNIT Surat (Gujarat) India for sup-porting this research paper Both authors carried out theproof All the authors conceived of the study and participatedin its design and coordination Both the authors read andapproved the final manuscript

References

[1] H H Khan ldquoOn the degree of approximation of a functionsbelonging to the class Lip(120572 p)rdquo Indian Journal of Pure andApplied Mathematics vol 5 pp 132ndash136 1974

[2] H H Khan Approximation of classes of functions [PhD thesis]AMU Aligarh India 1974

[3] V N Mishra and L N Mishra ldquoTrigonometric approximationin 119871119901(119901 ge 1) spacesrdquo International Journal of Contemporary

Mathematical Sciences vol 7 pp 909ndash918 2012[4] V N Mishra K Khatri and L N Mishra ldquoProduct

(119873 119901119899)(119862 1) summability of a sequence of Fourier coefficientsrdquo

Mathematical Sciences vol 6 p 38 2012[5] VNMishra KKhatri and LNMishra ldquoProduct summability

transform of conjugate series of Fourier seriesrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2012Article ID 298923 13 pages 2012

[6] V N Mishra H H Khan K Khatri and L N Mishra ldquoOnapproximation of conjugate of signals (Functions) belongingto the generalized weighted 119882(119871

119903 120585(119905)) (119903 ge 1)-class by

product summability means of conjugate series of fourier

International Journal of Mathematics and Mathematical Sciences 9

seriesrdquo International Journal of Mathematical Analysis vol 6no 35 pp 1703ndash1715 2012

[7] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to the weighted 119882(119871

119901 120585(119905)) (119901 ge 1)-

class by almost matrix summability method of its conjugateFourier seriesrdquo International Journal of Applied Mathematicsand Mechanics vol 5 no 7 pp 16ndash27 2009

[8] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to Generalized Weighted 119882(119871

119901 120585(119905))

(119901 ge 1)-class by product summability methodrdquo Journal ofInternational Academy of Physical Sciences vol 14 no 4 pp413ndash423 2010

[9] J T Chen and Y S Jeng ldquoDual series representation and itsapplications to a string subjected to support motionsrdquoAdvancesin Engineering Software vol 27 no 3 pp 227ndash238 1996

[10] J T Chen and H-K Hong ldquoReview of dual boundary elementmethods with emphasis on hypersingular integrals and diver-gent seriesrdquo Applied Mechanics Reviews vol 52 no 1 pp 17ndash321999

[11] J T Chen H-K Hong C S Yeh and S W Chyuan ldquoInte-gral representations and regularizations for a divergent seriessolution of a beam subjected to support motionsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 9 pp 909ndash925 1996

[12] M Mursaleen and S A Mohiuddine Convergence Methods ForDouble Sequences and Applications Springer 2014

[13] G Alexits Convergence Problems of Orthogonal Series Perga-mon Elmsford New York NY USA 1961

[14] P Chandra ldquoOn the degree of approximation of a class of func-tions by means of Fourier seriesrdquo Acta Mathematica Hungaricavol 52 no 3-4 pp 199ndash205 1988

[15] P Chandra ldquoA note on the degree of approximation of contin-uous functionsrdquo Acta Mathematica Hungarica vol 62 no 1-2pp 21ndash23 1993

[16] R NMohapatra and P Chandra ldquoHolder continuous functionsand their Euler Borel and Taylor meansrdquo Math Chronicle vol11 pp 81ndash96 1982

[17] R N Mohapatra and P Chandra ldquoDegree of approximation offunctions in the Hlder metricrdquo Acta Mathematica Hungaricavol 41 no 1-2 pp 67ndash76 1983

[18] B Szal ldquoOn the rate of strong summability by matrix means inthe generalized Holder metricrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 1 article 28 2008

[19] T Singh and P Mahajan ldquoError bound of periodic signals inthe Holder metricrdquo International Journal of Mathematics andMathematical Science vol 2008 Article ID 495075 9 pages2008

[20] S Lal and K N S Yadav ldquoOn degree of approximation offunction belonging to the Lipschitz class by (119862 1)(119864 1)meansof its Fourier seriesrdquo Bulletin of Calcutta Mathematical Societyvol 93 pp 191ndash196 2001

[21] E Z Psarakis and G V Moustakides ldquoAn L2-based method

for the design of 1-D zero phase FIR digital filtersrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 44 no 7 pp 591ndash601 1997

[22] G Das T Ghosh and B K Ray ldquoDegree of approximation offunctions in the Holder metric by (e c) Meansrdquo Proceedings ofthe Indian Academy of Science vol 105 pp 315ndash327 1995

[23] A ZygmundTrigonometric Series CambridgeUniversity PressLondon UK 3rd edition 2002

[24] G Bachman L Narici and E Beckenstien Fourier andWaveletAnalysis Springer New York NY USA 2000

[25] M I Gilrsquo ldquoEstimates for entries of matrix valued functions ofinfinite matricesrdquoMathematical Physics Analysis and Geometryvol 11 no 2 pp 175ndash186 2008

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi 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

International Journal of Mathematics and Mathematical Sciences 3

then the infinite seriessuminfin119899=0

119886119899is said to be (119873

119901sdot1198641

) summableto 119904

We note that 1198641119899 (119862119864)

1

119899 and

119873119864

119899are also trigonometric

polynomials of degree (or order) 119899The relation between Cesaro mean and Fejer mean is

given by

120590119899(119909) =

1

2120587

int

2120587

0

[119891 (119909 + 119905) + 119891 (119909 minus 119905)] 119865119899(119905) 119889119905 (19)

where the Fejer mean is

1198650(119905) = 1

119865119899(119905) =

1

119899 + 1

(

sin (119899 + 1) (1199052)sin (1199052)

)

2

119905 notin 2120587119885 119899 ge 1

(119899 + 1) 119905 isin 2120587119885

(20)

and Cesaro mean is

120590119899=

1199040(119909) + 119904

1(119909) + sdot sdot sdot + 119904

119899minus1(119909)

119899

=

1

119899

119899minus1

sum

119896=0

119904119896(119909) (21)

Some important particular cases are as follows

(1) (119867 1(119899 + 1)) sdot (1198641) means when 119886

119899119896= 1(119899 minus 119896 +

1) log 119899

(2) (119873 119901119899 119902119899) sdot 1198641means 119886

119899119896= 119901119899minus119896

119902119896119877119899 where 119877

119899=

sum119899

119896=0119901119896119902119899minus119896

= 0

Remark 1 If we take 120596(|119909 minus 119910|) = |119909 minus 119910|120572 then119867

120596reduces

to119867120572class

The conjugate function 119891(119909) is defined for almost every 119909

by

119891 (119909) = minus

1

2120587

int

120587

0

120595 (119905) cot 1199052

119889119905

= limℎrarr0

(minus

1

2120587

int

120587

ℎ

120595 (119905) cot 1199052

119889119905)

(22)

see [23 page 131]We write throughout the paper

120595119909(119905) = 120595 (119905) = 119891 (119909 + 119905) minus 119891 (119909 minus 119905)

120601119909(119905) = 119891 (119909 + 119905) minus 2119891 (119909) + 119891 (119909 minus 119905)

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(23)

We note that the series conjugate to a Fourier series is notnecessarily a Fourier series [23 24] Hence a separate studyof conjugate series is desirable and attracted the attention ofresearchers

2 Known Results

In 2001 Lal and Yadav [20] established the following theoremto estimate the error between the input signal 119891(119909) andthe signal obtained after passing through the (119862 1)(119864 1)-transform

Theorem 2 (see [20]) If a function 119891 119877 rarr 119877 is 2120587-periodicfunction and belongs to class Lip120572 0 lt 120572 le 1 then degreeof approximation by (119862 1)(119864 1) means of its Fourier series isgiven by

100381710038171003817100381710038171003817

1199051198621

sdot1198641

119899(119891) minus 119891 (119909)

100381710038171003817100381710038171003817infin

=

119874(119899minus120572

) 0 lt 120572 lt 1

119874(

log 119899119899

) 120572 = 1

(24)

Recently Singh and Mahajan [19] generalized the above resultunder more general assumptions They proved the following

Theorem 3 (see [19]) Let 120596(119905) defined in (5) be such that

int

120587

119905

120596 (119906)

1199062119889119906 = 119874 (119867 (119905)) 119867 (119905) ge 0

int

119905

0

119867(119906) 119889119906 = 119874 (119905119867 (119905)) as 119905 997888rarr 0+

(25)

then for 0 lt 120573 le 120572 le 1 and 119891 isin 119867120596 we have

100381710038171003817100381710038171003817

1199051198621

sdot1198641

119899(119891) minus 119891 (119909)

100381710038171003817100381710038171003817120596lowast

= 119874(((119899 + 1)minus1

119867(

120587

119899 + 1

))

1minus 120573120572

)

(26)

Theorem4 (see [19]) Consider120596(119905) defined in (5) and for 0 lt120573 le 120572 le 1 and 119891 isin 119867

120596 we have

100381710038171003817100381710038171003817

1199051198621

sdot1198641

119899(119891) minus 119891 (119909)

100381710038171003817100381710038171003817120596lowast

= 119874((120596(

120587

119899

+ 1))

1minus 120573120572

+((119899 + 1)minus1

119899+1

sum

119896=1

120596(

1

119896 + 1

))

1minus 120573120572

)

(27)

3 Main Theorem

In this paper we prove a theorem on the degree of approx-imation of a function

119891(119909) conjugate to a 2120587-periodicfunction 119891 belonging to

119891 isin 119867120596

class by (119873119901sdot 1198641

)

means of conjugate series of its Fourier series This workgeneralizes the results of Singh andMahajan [19] on (119873

119901sdot 1198641

)

summability of conjugate Fourier series We will measure theerror between the input signal 119891(119909) and the processed outputsignal 119873119864

119899(119891 119909) = (1119875

119899) sum119899

119896=0119901119899minus119896

1198641

119896(119891) by establishing the

following theorems

Theorem 5 The functions 120596 satisfy the following conditions

int

120587

119905

120596 (119906)

1199062119889119906 = 119874 (119867 (119905)) 119867 (119905) ge 0 (28)

int

119905

0

119867(119906) 119889119906 = 119874 (119905119867 (119905)) as 119905 997888rarr 0+

(29)

4 International Journal of Mathematics and Mathematical Sciences

where 120596(119905) and 120596lowast(119905) are increasing functions of 119905 Let 119873119901be

the Norlund summability matrix generated by the nonnegative119901119899 such that

(119899 + 1) 119901119899= 119874 (119875

119899) forall119899 ge 0 (30)

Then for 119891 isin 119867120596 0 lt 120573 le 120572 le 1 we have

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(31)

and if 120596(119905) satisfies (28) then for 119891 isin 119867

120596 0 lt 120573 le 120572 le 1 we

have10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(32)

Remark 6 The product transform (119873119901sdot 1198641

) plays an impor-tant role in signal theory as a double digital filter and theoryof Machines in Mechanical Engineering [4 5]

4 Lemmas

In order to prove our main Theorem 5 we require thefollowing lemmas

Lemma 7 If 120595119909(119905) = 120595(119905) = 119891(119909 + 119905) minus 119891(119909 minus 119905) then for

119891 isin 119867

120596 we have

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816le 2119872120596 (

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) (33)

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816le 2119872120596 (|119905|) (34)

It is easy to verify the following

Lemma 8 Let 119901119899 be a nonnegative and nonincreasing

sequence satisfies (30) then for 0 lt 119905 le 120587(119899 + 1) we have

(119873119901sdot 1198641

)119899

(119905) = 119874(

1

119905

) (35)

Proof Consider

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(36)

Using condition (30) sin 119899119905 le 119899119905 sin(1199052) ge (119905120587) for 0 lt

119905 le 120587 and |cos119896(1199052)| le 1 we have

(119873119901sdot 1198641

)119899

(119905)

le

119901119899

119875119899

119899

sum

119896=0

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

= 119874(

1

119905 (119899 + 1)

)[

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)]

= (

1

119905 (119899 + 1)

)

119899

sum

119896=0

Re cos119896 ( 1199052

) 119890119894(119896+1)1199052

= (

1

119905 (119899 + 1)

)Re1198901198941199052 (1 minus cos119899+1 (1199052) 119890 119894(119899+1)1199052

1 minus cos (1199052) 1198901198941199052)

= (

1

1199052(119899 + 1)

)Re 119894 minus 119894 cos (119899 + 1) (1199052) cos119899+1 (1199052)

+ sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

) sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

(119899 + 1) 119905)

= 119874(

1

119905

)

(37)

Lemma 9 Let 119901119899 be a nonnegative and nonincreasing

sequence satisfies (30) then for 120587(119899 + 1) le 119905 le 120587 we get

(119873119901sdot 1198641

)119899

(119905) = 119874(

1

1199052(119899 + 1)

) (38)

Proof Consider

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(39)

Using condition (30) sin (1199052) ge (119905120587) for 0 lt 119905 le 120587| sin 119905| le1 for all 119905 and |cos119896(1199052)| le 1 we have

(119873119901sdot 1198641

)119899

(119905)

le

119901119899

119875119899

119899

sum

119896=0

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

= (

1

119905 (119899 + 1)

)

119899

sum

119896=0

Re cos119896 ( 1199052

) 119890119894(119896+1)1199052

= (

1

119905 (119899 + 1)

)Re1198901198941199052 (1 minus cos119899+1 (1199052) 119890119894(119899+1)1199052

1 minus cos (1199052) 1198901198941199052)

International Journal of Mathematics and Mathematical Sciences 5

= 119874(

1

1199052(119899 + 1)

) sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

)

(40)

Lemma 10 (see [19]) If120596(119905) satisfies conditions (28) and (29)then

int

119906

0

119905minus1

120596 (119905) 119889119905 = 119874 (119906119867 (119906)) as 119906 997888rarr 0+

(41)

5 Proof of Theorem 5

The integral representationof 119904119899(119891 119909) is given by

119904119899(119891 119909) =

infin

sum

119899=1

(119887119899cos 119899119909 minus 119886

119899sin 119899119909)

= minus

1

120587

int

120587

0

120595119909(119905) (

cot (1199052) cos (119899 + 12) 1199052 sin (1199052)

) 119889119905

(42)

and therefore

119904119899(119891 119909) minus

119891 (119909) =

1

2120587

int

120587

0

120595119909(119905)

cos (119899 + 12) 119905sin 1199052

119889119905 (43)

Denoting 1198641 means of 119904119899(119891 119909) by 1198641

119899(119909) we have

1

2119899

119899

sum

119896=0

(

119899

119896) 119904119899(119891 119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

1198641

119899(119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re

119899

sum

119896=0

(

119899

119896) 119890119894(119896+12)119905

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 1198901198941199052(1 + 119890119894119905)

119899

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 2119899cos119899 ( 119905

2

) 119890119894(119899+1)1199052

119889119905

=

1

2120587

int

120587

0

120595119909(119905)

cos119899 (1199052) cos (119899 + 1) (1199052)sin 1199052

119889119905

(44)

Now using condition (30) then 119873119901transform of 1198641 trans-

form is given by

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

=

1

2120587

int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

119901119899minus119896

119875119899

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

)int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

(45)

Set

119864119899(119909) =

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

= 119874(

1

119899 + 1

)

times int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

119864119899(119909 119910)

=1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

= 119874(

1

119899 + 1

)int

120587

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

) [int

120587(119899+1)

0

+int

120587

120587(119899+1)

]

= 1198681+ 1198682 (say)

(46)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198681= 119874 (1) int

120587(119899+1)

0

119905minus1

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(47)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(48)

6 International Journal of Mathematics and Mathematical Sciences

Now using (33) Lemma 8 we get

1198681= 119874(

1

119899 + 1

) int

120587(119899+1)

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

119889119905

= 119874(

1

119899 + 1

) int

120587(119899+1)

0

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

119905

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874 (log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(49)

Now using (33) Lemma 9 we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) 119889119905

= 119874(

1

119899 + 1

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))int

120587

120587(119899+1)

119905minus2

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(50)

Using the fact that we may write 119868119896= 1198681minus120573120572

119896119868120573120572

119896 119896 = 1 2

and combining (47) and (49) we get

1198681= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(51)

Combining (48) and (50) we get

1198682= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(52)

Therefore from (8) (51) and (52) we have

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= sup119909119910

1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(53)

Since1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= sup0le119909le2120587

100381610038161003816100381610038161003816

119879sdot1198641

119899(119891) minus

119891 (119909)

100381610038161003816100381610038161003816

(54)

it follows from (47) and (48) that

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874((119899 + 1)

minus1

119867(

120587

119899 + 1

)) (55)

Combining (53) and (55) we get

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times[(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(56)

To prove (32) if 120596(119905) satisfies only (28) then using (34) andsecond mean value theorem for integrals we have

1198681= 119874 (120596 (120587 (119899 + 1))) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

(57)

Combining (49) and (57) we get

1198681= 119874([log (119899 + 1) 120596 ( 120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(58)

Again if 120596(119905) satisfies only (28) then using (34) and secondmean value theorem for integrals we have

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

120596 (119905)

1199052119889119905

= 119874(

1

119899 + 1

)int

119899+1

1

120596(

120587

119905

) 119889119905

= 119874(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)

(59)

Combining (50) and (59) we get

1198682= 119874([(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(60)

International Journal of Mathematics and Mathematical Sciences 7

Therefore

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times ( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(61)

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

+ 119874((

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

))

(62)

Combining (61) and (62) we get10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times (log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(63)

This completes the proof of Theorem 5

6 Applications

The theory of approximation is a very extensive field andthe study of theory of trigonometric approximation is ofgreatmathematical interest and of great practical importanceAs mentioned in [21] the 119871

119901space in general and 119871

2and

119871infin

in particular play an important role in the theory ofsignals and filters From the point of view of the applicationssharper estimates of infinite matrices [25] are useful to getbounds for the lattice norms (which occur in solid statephysics) of matrix valued functions and enable to investigateperturbations of matrix valued functions and compare themThe following corollaries may be derived fromTheorem 5

If 120596(|119909 minus 119910|) le 119861|119909 minus 119910|120572

120596lowast

(|119909 minus 119910|) le 119870|119909 minus 119910|120573

0 le

120573 lt 120572 le 1 and set

119867(119906) =

119906120572minus1

0 lt 120572 lt 1

log(1119906

) 120572 = 1

(64)

then we get Corollary 11

Corollary 11 If 119891 isin 119867120572 0 lt 120573 le 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120573

=

119874((log (119899 + 1))120573120572(119899 + 1)120573minus120572) 0 lt 120572 lt 1

119874 (log (119899 + 1) (119899 + 1)120573minus1) 120572 = 1

(65)

If we put 120573 = 0 in Corollary 11 then we get Corollary 12

Corollary 12 If 119891 isin Lip120572 0 lt 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817119888=

119874((119899 + 1)minus120572

) 0 lt 120572 lt 1

119874(

log (119899 + 1)(119899 + 1)

) 120572 = 1

(66)

7 Example

In the example we see how the sequence of averages (ie1205901

119899(119909)-means or (119862 1)mean) andNorlundmean119873

119901of partial

sums of a Fourier series is better behaved than the sequenceof partial sums 119904

119899(119909) itself

Let

119891 (119909) =

minus1 minus120587 le 119909 lt 0

1 0 le 119909 lt 120587

(67)

with 119891(119909 + 2120587) = 119891(119909) for all real 119909 Fourier series of 119891(119909) isgiven by

2

120587

infin

sum

119899=1

1 minus (minus1)119899

119899

sin 119899119909 minus120587 le 119909 le 120587 (68)

Then 119899th partial sum 119904119899(119909) of Fourier series (68) and 119899th

Cesaro sum for 120575 = 1 that is 1205901119899(119909) for the series (68) are

given by

119904119899(119909) =

4

120587

(sin119909 + 1

3

sin 3119909 + sdot sdot sdot + 1

119899

sin 119899119909)

1205901

119899(119909) =

2

120587

119899

sum

119896=1

(1 minus

119896

119899

)(

1 minus (minus1)119896

119896

) sin 119896119909(69)

FromTheorem 20 of Hardyrsquos ldquoDivergent Seriesrdquo if a Norlundmethod 119873

119901has increasing weights 119901

119899 then it is stronger

than (119862 1)Now take119873

119901to be the Norlundmatrix generated by 119901

119899=

119899 + 1 then Norlund means119873119901is given by

119905119873

119899(119891 119909) =

2

(119899 + 1) (119899 + 2)

119899

sum

119896=0

(119899 minus 119896 + 1) 119904119896(119891 119909) (70)

In this graph we observe that 1205901119899(119909) 119905119873

119899(119891 119909) converges to

119891(119909) faster than 119904119899(119909) in the interval [minus120587 120587] We further note

that near the points of discontinuities that is minus120587 0 and 120587the graph of 119904

5and 11990410

shows peaks that move closer to theline passing through points of discontinuity as n increases(Gibbs phenomenon) but in the graph of 1205901

119899(119909) 119905119873

119899(119891 119909)

8 International Journal of Mathematics and Mathematical Sciences

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(a)

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(b)

Figure 1 Graph of 119891(119909) (blue) 119904119899(119909) (pink) 1205901

119899(119909) (yellow) 119905119873

119899(119891 119909) (green) 119899 = 5 and 10

119899 = 5 10 the peaks becomeflatter (Figure 1)TheGibbs phe-nomenon is an overshoot a peculiarity of the Fourier seriesand other eigen function series at a simple discontinuitythat is the convergence of Fourier series is very slow nearthe point of discontinuity Thus the product summabilitymeans of the Fourier series of 119891(119909) overshoot the GibbsPhenomenon and show the smoothing effect of the methodThus 1205901

119899(119909) 119905119873

119899(119891 119909) is the better approximant than 119904

119899(119909) and

119873119901method is stronger than (119862 1) method

8 Conclusion

Several results concerning the degree of approximation ofperiodic signals (functions) by product summability meansof Fourier series and conjugate Fourier series in generalizedHolder metric and Holder metric have been reviewed Usinggraphical representation 120590

1

119899(119909) 119905119873

119899(119891 119909) is a better approxi-

mant to 119904119899(119909) but till now nothing has been done to show

this Some interesting application of the operator (119873119901sdot 1198641

)

used in this paper is pointed out in Remark 6 Also the resultof our theorem is more general rather than the results of anyother previous proved theorems whichwill enrich the literateof summability theory of infinite series

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude tothe anonymous learned referee(s) and the editor for theirvaluable suggestions and constructive comments whichresulted in the subsequent improvement of this paper Specialthanks are due to Professor Ricardo Estrada Academic Editorof IJMMS and Professor Mohammed Mahmoud Editorial

office Hindawi Publishing Corporation for kind cooper-ation smooth behavior during communication and theirefforts to send the reports of the paper timelyThe authors arealso grateful to all the editorial boardmembers and reviewersof Hindawi prestigious journal that is International Journalof Mathematics and Mathematical Sciences The secondauthor Kejal Khatri acknowledges the Ministry of HumanResource Development New Delhi India for providing thefinancial support and to carry out her research work underthe guidance of Dr Vishnu Narayan Mishra at SVNITSurat (Gujarat) India The first author Vishnu NarayanMishra acknowledges Cumulative Professional DevelopmentAllowance (CPDA) SVNIT Surat (Gujarat) India for sup-porting this research paper Both authors carried out theproof All the authors conceived of the study and participatedin its design and coordination Both the authors read andapproved the final manuscript

References

[1] H H Khan ldquoOn the degree of approximation of a functionsbelonging to the class Lip(120572 p)rdquo Indian Journal of Pure andApplied Mathematics vol 5 pp 132ndash136 1974

[2] H H Khan Approximation of classes of functions [PhD thesis]AMU Aligarh India 1974

[3] V N Mishra and L N Mishra ldquoTrigonometric approximationin 119871119901(119901 ge 1) spacesrdquo International Journal of Contemporary

Mathematical Sciences vol 7 pp 909ndash918 2012[4] V N Mishra K Khatri and L N Mishra ldquoProduct

(119873 119901119899)(119862 1) summability of a sequence of Fourier coefficientsrdquo

Mathematical Sciences vol 6 p 38 2012[5] VNMishra KKhatri and LNMishra ldquoProduct summability

transform of conjugate series of Fourier seriesrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2012Article ID 298923 13 pages 2012

[6] V N Mishra H H Khan K Khatri and L N Mishra ldquoOnapproximation of conjugate of signals (Functions) belongingto the generalized weighted 119882(119871

119903 120585(119905)) (119903 ge 1)-class by

product summability means of conjugate series of fourier

International Journal of Mathematics and Mathematical Sciences 9

seriesrdquo International Journal of Mathematical Analysis vol 6no 35 pp 1703ndash1715 2012

[7] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to the weighted 119882(119871

119901 120585(119905)) (119901 ge 1)-

class by almost matrix summability method of its conjugateFourier seriesrdquo International Journal of Applied Mathematicsand Mechanics vol 5 no 7 pp 16ndash27 2009

[8] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to Generalized Weighted 119882(119871

119901 120585(119905))

(119901 ge 1)-class by product summability methodrdquo Journal ofInternational Academy of Physical Sciences vol 14 no 4 pp413ndash423 2010

[9] J T Chen and Y S Jeng ldquoDual series representation and itsapplications to a string subjected to support motionsrdquoAdvancesin Engineering Software vol 27 no 3 pp 227ndash238 1996

[10] J T Chen and H-K Hong ldquoReview of dual boundary elementmethods with emphasis on hypersingular integrals and diver-gent seriesrdquo Applied Mechanics Reviews vol 52 no 1 pp 17ndash321999

[11] J T Chen H-K Hong C S Yeh and S W Chyuan ldquoInte-gral representations and regularizations for a divergent seriessolution of a beam subjected to support motionsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 9 pp 909ndash925 1996

[12] M Mursaleen and S A Mohiuddine Convergence Methods ForDouble Sequences and Applications Springer 2014

[13] G Alexits Convergence Problems of Orthogonal Series Perga-mon Elmsford New York NY USA 1961

[14] P Chandra ldquoOn the degree of approximation of a class of func-tions by means of Fourier seriesrdquo Acta Mathematica Hungaricavol 52 no 3-4 pp 199ndash205 1988

[15] P Chandra ldquoA note on the degree of approximation of contin-uous functionsrdquo Acta Mathematica Hungarica vol 62 no 1-2pp 21ndash23 1993

[16] R NMohapatra and P Chandra ldquoHolder continuous functionsand their Euler Borel and Taylor meansrdquo Math Chronicle vol11 pp 81ndash96 1982

[17] R N Mohapatra and P Chandra ldquoDegree of approximation offunctions in the Hlder metricrdquo Acta Mathematica Hungaricavol 41 no 1-2 pp 67ndash76 1983

[18] B Szal ldquoOn the rate of strong summability by matrix means inthe generalized Holder metricrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 1 article 28 2008

[19] T Singh and P Mahajan ldquoError bound of periodic signals inthe Holder metricrdquo International Journal of Mathematics andMathematical Science vol 2008 Article ID 495075 9 pages2008

[20] S Lal and K N S Yadav ldquoOn degree of approximation offunction belonging to the Lipschitz class by (119862 1)(119864 1)meansof its Fourier seriesrdquo Bulletin of Calcutta Mathematical Societyvol 93 pp 191ndash196 2001

[21] E Z Psarakis and G V Moustakides ldquoAn L2-based method

for the design of 1-D zero phase FIR digital filtersrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 44 no 7 pp 591ndash601 1997

[22] G Das T Ghosh and B K Ray ldquoDegree of approximation offunctions in the Holder metric by (e c) Meansrdquo Proceedings ofthe Indian Academy of Science vol 105 pp 315ndash327 1995

[23] A ZygmundTrigonometric Series CambridgeUniversity PressLondon UK 3rd edition 2002

[24] G Bachman L Narici and E Beckenstien Fourier andWaveletAnalysis Springer New York NY USA 2000

[25] M I Gilrsquo ldquoEstimates for entries of matrix valued functions ofinfinite matricesrdquoMathematical Physics Analysis and Geometryvol 11 no 2 pp 175ndash186 2008

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

4 International Journal of Mathematics and Mathematical Sciences

where 120596(119905) and 120596lowast(119905) are increasing functions of 119905 Let 119873119901be

the Norlund summability matrix generated by the nonnegative119901119899 such that

(119899 + 1) 119901119899= 119874 (119875

119899) forall119899 ge 0 (30)

Then for 119891 isin 119867120596 0 lt 120573 le 120572 le 1 we have

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(31)

and if 120596(119905) satisfies (28) then for 119891 isin 119867

120596 0 lt 120573 le 120572 le 1 we

have10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(32)

Remark 6 The product transform (119873119901sdot 1198641

) plays an impor-tant role in signal theory as a double digital filter and theoryof Machines in Mechanical Engineering [4 5]

4 Lemmas

In order to prove our main Theorem 5 we require thefollowing lemmas

Lemma 7 If 120595119909(119905) = 120595(119905) = 119891(119909 + 119905) minus 119891(119909 minus 119905) then for

119891 isin 119867

120596 we have

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816le 2119872120596 (

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) (33)

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816le 2119872120596 (|119905|) (34)

It is easy to verify the following

Lemma 8 Let 119901119899 be a nonnegative and nonincreasing

sequence satisfies (30) then for 0 lt 119905 le 120587(119899 + 1) we have

(119873119901sdot 1198641

)119899

(119905) = 119874(

1

119905

) (35)

Proof Consider

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(36)

Using condition (30) sin 119899119905 le 119899119905 sin(1199052) ge (119905120587) for 0 lt

119905 le 120587 and |cos119896(1199052)| le 1 we have

(119873119901sdot 1198641

)119899

(119905)

le

119901119899

119875119899

119899

sum

119896=0

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

= 119874(

1

119905 (119899 + 1)

)[

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)]

= (

1

119905 (119899 + 1)

)

119899

sum

119896=0

Re cos119896 ( 1199052

) 119890119894(119896+1)1199052

= (

1

119905 (119899 + 1)

)Re1198901198941199052 (1 minus cos119899+1 (1199052) 119890 119894(119899+1)1199052

1 minus cos (1199052) 1198901198941199052)

= (

1

1199052(119899 + 1)

)Re 119894 minus 119894 cos (119899 + 1) (1199052) cos119899+1 (1199052)

+ sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

) sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

(119899 + 1) 119905)

= 119874(

1

119905

)

(37)

Lemma 9 Let 119901119899 be a nonnegative and nonincreasing

sequence satisfies (30) then for 120587(119899 + 1) le 119905 le 120587 we get

(119873119901sdot 1198641

)119899

(119905) = 119874(

1

1199052(119899 + 1)

) (38)

Proof Consider

(119873119901sdot 1198641

)119899

(119905) =

1

119875119899

119899

sum

119896=0

119901119899minus119896

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

(39)

Using condition (30) sin (1199052) ge (119905120587) for 0 lt 119905 le 120587| sin 119905| le1 for all 119905 and |cos119896(1199052)| le 1 we have

(119873119901sdot 1198641

)119899

(119905)

le

119901119899

119875119899

119899

sum

119896=0

cos119896 (1199052) cos (119896 + 1) (1199052)sin 1199052

= (

1

119905 (119899 + 1)

)

119899

sum

119896=0

Re cos119896 ( 1199052

) 119890119894(119896+1)1199052

= (

1

119905 (119899 + 1)

)Re1198901198941199052 (1 minus cos119899+1 (1199052) 119890119894(119899+1)1199052

1 minus cos (1199052) 1198901198941199052)

International Journal of Mathematics and Mathematical Sciences 5

= 119874(

1

1199052(119899 + 1)

) sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

)

(40)

Lemma 10 (see [19]) If120596(119905) satisfies conditions (28) and (29)then

int

119906

0

119905minus1

120596 (119905) 119889119905 = 119874 (119906119867 (119906)) as 119906 997888rarr 0+

(41)

5 Proof of Theorem 5

The integral representationof 119904119899(119891 119909) is given by

119904119899(119891 119909) =

infin

sum

119899=1

(119887119899cos 119899119909 minus 119886

119899sin 119899119909)

= minus

1

120587

int

120587

0

120595119909(119905) (

cot (1199052) cos (119899 + 12) 1199052 sin (1199052)

) 119889119905

(42)

and therefore

119904119899(119891 119909) minus

119891 (119909) =

1

2120587

int

120587

0

120595119909(119905)

cos (119899 + 12) 119905sin 1199052

119889119905 (43)

Denoting 1198641 means of 119904119899(119891 119909) by 1198641

119899(119909) we have

1

2119899

119899

sum

119896=0

(

119899

119896) 119904119899(119891 119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

1198641

119899(119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re

119899

sum

119896=0

(

119899

119896) 119890119894(119896+12)119905

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 1198901198941199052(1 + 119890119894119905)

119899

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 2119899cos119899 ( 119905

2

) 119890119894(119899+1)1199052

119889119905

=

1

2120587

int

120587

0

120595119909(119905)

cos119899 (1199052) cos (119899 + 1) (1199052)sin 1199052

119889119905

(44)

Now using condition (30) then 119873119901transform of 1198641 trans-

form is given by

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

=

1

2120587

int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

119901119899minus119896

119875119899

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

)int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

(45)

Set

119864119899(119909) =

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

= 119874(

1

119899 + 1

)

times int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

119864119899(119909 119910)

=1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

= 119874(

1

119899 + 1

)int

120587

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

) [int

120587(119899+1)

0

+int

120587

120587(119899+1)

]

= 1198681+ 1198682 (say)

(46)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198681= 119874 (1) int

120587(119899+1)

0

119905minus1

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(47)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(48)

6 International Journal of Mathematics and Mathematical Sciences

Now using (33) Lemma 8 we get

1198681= 119874(

1

119899 + 1

) int

120587(119899+1)

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

119889119905

= 119874(

1

119899 + 1

) int

120587(119899+1)

0

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

119905

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874 (log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(49)

Now using (33) Lemma 9 we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) 119889119905

= 119874(

1

119899 + 1

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))int

120587

120587(119899+1)

119905minus2

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(50)

Using the fact that we may write 119868119896= 1198681minus120573120572

119896119868120573120572

119896 119896 = 1 2

and combining (47) and (49) we get

1198681= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(51)

Combining (48) and (50) we get

1198682= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(52)

Therefore from (8) (51) and (52) we have

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= sup119909119910

1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(53)

Since1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= sup0le119909le2120587

100381610038161003816100381610038161003816

119879sdot1198641

119899(119891) minus

119891 (119909)

100381610038161003816100381610038161003816

(54)

it follows from (47) and (48) that

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874((119899 + 1)

minus1

119867(

120587

119899 + 1

)) (55)

Combining (53) and (55) we get

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times[(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(56)

To prove (32) if 120596(119905) satisfies only (28) then using (34) andsecond mean value theorem for integrals we have

1198681= 119874 (120596 (120587 (119899 + 1))) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

(57)

Combining (49) and (57) we get

1198681= 119874([log (119899 + 1) 120596 ( 120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(58)

Again if 120596(119905) satisfies only (28) then using (34) and secondmean value theorem for integrals we have

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

120596 (119905)

1199052119889119905

= 119874(

1

119899 + 1

)int

119899+1

1

120596(

120587

119905

) 119889119905

= 119874(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)

(59)

Combining (50) and (59) we get

1198682= 119874([(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(60)

International Journal of Mathematics and Mathematical Sciences 7

Therefore

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times ( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(61)

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

+ 119874((

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

))

(62)

Combining (61) and (62) we get10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times (log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(63)

This completes the proof of Theorem 5

6 Applications

The theory of approximation is a very extensive field andthe study of theory of trigonometric approximation is ofgreatmathematical interest and of great practical importanceAs mentioned in [21] the 119871

119901space in general and 119871

2and

119871infin

in particular play an important role in the theory ofsignals and filters From the point of view of the applicationssharper estimates of infinite matrices [25] are useful to getbounds for the lattice norms (which occur in solid statephysics) of matrix valued functions and enable to investigateperturbations of matrix valued functions and compare themThe following corollaries may be derived fromTheorem 5

If 120596(|119909 minus 119910|) le 119861|119909 minus 119910|120572

120596lowast

(|119909 minus 119910|) le 119870|119909 minus 119910|120573

0 le

120573 lt 120572 le 1 and set

119867(119906) =

119906120572minus1

0 lt 120572 lt 1

log(1119906

) 120572 = 1

(64)

then we get Corollary 11

Corollary 11 If 119891 isin 119867120572 0 lt 120573 le 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120573

=

119874((log (119899 + 1))120573120572(119899 + 1)120573minus120572) 0 lt 120572 lt 1

119874 (log (119899 + 1) (119899 + 1)120573minus1) 120572 = 1

(65)

If we put 120573 = 0 in Corollary 11 then we get Corollary 12

Corollary 12 If 119891 isin Lip120572 0 lt 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817119888=

119874((119899 + 1)minus120572

) 0 lt 120572 lt 1

119874(

log (119899 + 1)(119899 + 1)

) 120572 = 1

(66)

7 Example

In the example we see how the sequence of averages (ie1205901

119899(119909)-means or (119862 1)mean) andNorlundmean119873

119901of partial

sums of a Fourier series is better behaved than the sequenceof partial sums 119904

119899(119909) itself

Let

119891 (119909) =

minus1 minus120587 le 119909 lt 0

1 0 le 119909 lt 120587

(67)

with 119891(119909 + 2120587) = 119891(119909) for all real 119909 Fourier series of 119891(119909) isgiven by

2

120587

infin

sum

119899=1

1 minus (minus1)119899

119899

sin 119899119909 minus120587 le 119909 le 120587 (68)

Then 119899th partial sum 119904119899(119909) of Fourier series (68) and 119899th

Cesaro sum for 120575 = 1 that is 1205901119899(119909) for the series (68) are

given by

119904119899(119909) =

4

120587

(sin119909 + 1

3

sin 3119909 + sdot sdot sdot + 1

119899

sin 119899119909)

1205901

119899(119909) =

2

120587

119899

sum

119896=1

(1 minus

119896

119899

)(

1 minus (minus1)119896

119896

) sin 119896119909(69)

FromTheorem 20 of Hardyrsquos ldquoDivergent Seriesrdquo if a Norlundmethod 119873

119901has increasing weights 119901

119899 then it is stronger

than (119862 1)Now take119873

119901to be the Norlundmatrix generated by 119901

119899=

119899 + 1 then Norlund means119873119901is given by

119905119873

119899(119891 119909) =

2

(119899 + 1) (119899 + 2)

119899

sum

119896=0

(119899 minus 119896 + 1) 119904119896(119891 119909) (70)

In this graph we observe that 1205901119899(119909) 119905119873

119899(119891 119909) converges to

119891(119909) faster than 119904119899(119909) in the interval [minus120587 120587] We further note

that near the points of discontinuities that is minus120587 0 and 120587the graph of 119904

5and 11990410

shows peaks that move closer to theline passing through points of discontinuity as n increases(Gibbs phenomenon) but in the graph of 1205901

119899(119909) 119905119873

119899(119891 119909)

8 International Journal of Mathematics and Mathematical Sciences

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(a)

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(b)

Figure 1 Graph of 119891(119909) (blue) 119904119899(119909) (pink) 1205901

119899(119909) (yellow) 119905119873

119899(119891 119909) (green) 119899 = 5 and 10

119899 = 5 10 the peaks becomeflatter (Figure 1)TheGibbs phe-nomenon is an overshoot a peculiarity of the Fourier seriesand other eigen function series at a simple discontinuitythat is the convergence of Fourier series is very slow nearthe point of discontinuity Thus the product summabilitymeans of the Fourier series of 119891(119909) overshoot the GibbsPhenomenon and show the smoothing effect of the methodThus 1205901

119899(119909) 119905119873

119899(119891 119909) is the better approximant than 119904

119899(119909) and

119873119901method is stronger than (119862 1) method

8 Conclusion

Several results concerning the degree of approximation ofperiodic signals (functions) by product summability meansof Fourier series and conjugate Fourier series in generalizedHolder metric and Holder metric have been reviewed Usinggraphical representation 120590

1

119899(119909) 119905119873

119899(119891 119909) is a better approxi-

mant to 119904119899(119909) but till now nothing has been done to show

this Some interesting application of the operator (119873119901sdot 1198641

)

used in this paper is pointed out in Remark 6 Also the resultof our theorem is more general rather than the results of anyother previous proved theorems whichwill enrich the literateof summability theory of infinite series

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude tothe anonymous learned referee(s) and the editor for theirvaluable suggestions and constructive comments whichresulted in the subsequent improvement of this paper Specialthanks are due to Professor Ricardo Estrada Academic Editorof IJMMS and Professor Mohammed Mahmoud Editorial

office Hindawi Publishing Corporation for kind cooper-ation smooth behavior during communication and theirefforts to send the reports of the paper timelyThe authors arealso grateful to all the editorial boardmembers and reviewersof Hindawi prestigious journal that is International Journalof Mathematics and Mathematical Sciences The secondauthor Kejal Khatri acknowledges the Ministry of HumanResource Development New Delhi India for providing thefinancial support and to carry out her research work underthe guidance of Dr Vishnu Narayan Mishra at SVNITSurat (Gujarat) India The first author Vishnu NarayanMishra acknowledges Cumulative Professional DevelopmentAllowance (CPDA) SVNIT Surat (Gujarat) India for sup-porting this research paper Both authors carried out theproof All the authors conceived of the study and participatedin its design and coordination Both the authors read andapproved the final manuscript

References

[1] H H Khan ldquoOn the degree of approximation of a functionsbelonging to the class Lip(120572 p)rdquo Indian Journal of Pure andApplied Mathematics vol 5 pp 132ndash136 1974

[2] H H Khan Approximation of classes of functions [PhD thesis]AMU Aligarh India 1974

[3] V N Mishra and L N Mishra ldquoTrigonometric approximationin 119871119901(119901 ge 1) spacesrdquo International Journal of Contemporary

Mathematical Sciences vol 7 pp 909ndash918 2012[4] V N Mishra K Khatri and L N Mishra ldquoProduct

(119873 119901119899)(119862 1) summability of a sequence of Fourier coefficientsrdquo

Mathematical Sciences vol 6 p 38 2012[5] VNMishra KKhatri and LNMishra ldquoProduct summability

transform of conjugate series of Fourier seriesrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2012Article ID 298923 13 pages 2012

[6] V N Mishra H H Khan K Khatri and L N Mishra ldquoOnapproximation of conjugate of signals (Functions) belongingto the generalized weighted 119882(119871

119903 120585(119905)) (119903 ge 1)-class by

product summability means of conjugate series of fourier

International Journal of Mathematics and Mathematical Sciences 9

seriesrdquo International Journal of Mathematical Analysis vol 6no 35 pp 1703ndash1715 2012

[7] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to the weighted 119882(119871

119901 120585(119905)) (119901 ge 1)-

class by almost matrix summability method of its conjugateFourier seriesrdquo International Journal of Applied Mathematicsand Mechanics vol 5 no 7 pp 16ndash27 2009

[8] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to Generalized Weighted 119882(119871

119901 120585(119905))

(119901 ge 1)-class by product summability methodrdquo Journal ofInternational Academy of Physical Sciences vol 14 no 4 pp413ndash423 2010

[9] J T Chen and Y S Jeng ldquoDual series representation and itsapplications to a string subjected to support motionsrdquoAdvancesin Engineering Software vol 27 no 3 pp 227ndash238 1996

[10] J T Chen and H-K Hong ldquoReview of dual boundary elementmethods with emphasis on hypersingular integrals and diver-gent seriesrdquo Applied Mechanics Reviews vol 52 no 1 pp 17ndash321999

[11] J T Chen H-K Hong C S Yeh and S W Chyuan ldquoInte-gral representations and regularizations for a divergent seriessolution of a beam subjected to support motionsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 9 pp 909ndash925 1996

[12] M Mursaleen and S A Mohiuddine Convergence Methods ForDouble Sequences and Applications Springer 2014

[13] G Alexits Convergence Problems of Orthogonal Series Perga-mon Elmsford New York NY USA 1961

[14] P Chandra ldquoOn the degree of approximation of a class of func-tions by means of Fourier seriesrdquo Acta Mathematica Hungaricavol 52 no 3-4 pp 199ndash205 1988

[15] P Chandra ldquoA note on the degree of approximation of contin-uous functionsrdquo Acta Mathematica Hungarica vol 62 no 1-2pp 21ndash23 1993

[16] R NMohapatra and P Chandra ldquoHolder continuous functionsand their Euler Borel and Taylor meansrdquo Math Chronicle vol11 pp 81ndash96 1982

[17] R N Mohapatra and P Chandra ldquoDegree of approximation offunctions in the Hlder metricrdquo Acta Mathematica Hungaricavol 41 no 1-2 pp 67ndash76 1983

[18] B Szal ldquoOn the rate of strong summability by matrix means inthe generalized Holder metricrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 1 article 28 2008

[19] T Singh and P Mahajan ldquoError bound of periodic signals inthe Holder metricrdquo International Journal of Mathematics andMathematical Science vol 2008 Article ID 495075 9 pages2008

[20] S Lal and K N S Yadav ldquoOn degree of approximation offunction belonging to the Lipschitz class by (119862 1)(119864 1)meansof its Fourier seriesrdquo Bulletin of Calcutta Mathematical Societyvol 93 pp 191ndash196 2001

[21] E Z Psarakis and G V Moustakides ldquoAn L2-based method

for the design of 1-D zero phase FIR digital filtersrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 44 no 7 pp 591ndash601 1997

[22] G Das T Ghosh and B K Ray ldquoDegree of approximation offunctions in the Holder metric by (e c) Meansrdquo Proceedings ofthe Indian Academy of Science vol 105 pp 315ndash327 1995

[23] A ZygmundTrigonometric Series CambridgeUniversity PressLondon UK 3rd edition 2002

[24] G Bachman L Narici and E Beckenstien Fourier andWaveletAnalysis Springer New York NY USA 2000

[25] M I Gilrsquo ldquoEstimates for entries of matrix valued functions ofinfinite matricesrdquoMathematical Physics Analysis and Geometryvol 11 no 2 pp 175ndash186 2008

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

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Stochastic AnalysisInternational Journal of

International Journal of Mathematics and Mathematical Sciences 5

= 119874(

1

1199052(119899 + 1)

) sin (119899 + 1) (1199052) cos119899+1 (1199052)

= 119874(

1

1199052(119899 + 1)

)

(40)

Lemma 10 (see [19]) If120596(119905) satisfies conditions (28) and (29)then

int

119906

0

119905minus1

120596 (119905) 119889119905 = 119874 (119906119867 (119906)) as 119906 997888rarr 0+

(41)

5 Proof of Theorem 5

The integral representationof 119904119899(119891 119909) is given by

119904119899(119891 119909) =

infin

sum

119899=1

(119887119899cos 119899119909 minus 119886

119899sin 119899119909)

= minus

1

120587

int

120587

0

120595119909(119905) (

cot (1199052) cos (119899 + 12) 1199052 sin (1199052)

) 119889119905

(42)

and therefore

119904119899(119891 119909) minus

119891 (119909) =

1

2120587

int

120587

0

120595119909(119905)

cos (119899 + 12) 119905sin 1199052

119889119905 (43)

Denoting 1198641 means of 119904119899(119891 119909) by 1198641

119899(119909) we have

1

2119899

119899

sum

119896=0

(

119899

119896) 119904119899(119891 119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

1198641

119899(119909) minus

119891 (119909)

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052

119899

sum

119896=0

(

119899

119896) cos (119896 + 12) 119905 119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re

119899

sum

119896=0

(

119899

119896) 119890119894(119896+12)119905

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 1198901198941199052(1 + 119890119894119905)

119899

119889119905

=

1

2(119899+1)

120587

int

120587

0

120595119909(119905)

sin 1199052Re 2119899cos119899 ( 119905

2

) 119890119894(119899+1)1199052

119889119905

=

1

2120587

int

120587

0

120595119909(119905)

cos119899 (1199052) cos (119899 + 1) (1199052)sin 1199052

119889119905

(44)

Now using condition (30) then 119873119901transform of 1198641 trans-

form is given by

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

=

1

2120587

int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

119901119899minus119896

119875119899

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

)int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

(45)

Set

119864119899(119909) =

10038161003816100381610038161003816119873119864

119899(119891) minus

119891 (119909)

10038161003816100381610038161003816

= 119874(

1

119899 + 1

)

times int

120587

0

1003816100381610038161003816120595119909(119905)1003816100381610038161003816

119905

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

119864119899(119909 119910)

=1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

= 119874(

1

119899 + 1

)int

120587

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

times

1003816100381610038161003816100381610038161003816100381610038161003816

119899

sum

119896=0

cos119896 ( 1199052

) cos (119896 + 1) ( 1199052

)

1003816100381610038161003816100381610038161003816100381610038161003816

119889119905

= 119874(

1

119899 + 1

) [int

120587(119899+1)

0

+int

120587

120587(119899+1)

]

= 1198681+ 1198682 (say)

(46)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198681= 119874 (1) int

120587(119899+1)

0

119905minus1

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(47)

Now using (34) and Lemma 10 assume that120596(119905) satisfies (28)and (29) we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (119905) 119889119905

= 119874((119899 + 1)minus1

119867(

120587

119899 + 1

))

(48)

6 International Journal of Mathematics and Mathematical Sciences

Now using (33) Lemma 8 we get

1198681= 119874(

1

119899 + 1

) int

120587(119899+1)

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

119889119905

= 119874(

1

119899 + 1

) int

120587(119899+1)

0

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

119905

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874 (log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(49)

Now using (33) Lemma 9 we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) 119889119905

= 119874(

1

119899 + 1

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))int

120587

120587(119899+1)

119905minus2

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(50)

Using the fact that we may write 119868119896= 1198681minus120573120572

119896119868120573120572

119896 119896 = 1 2

and combining (47) and (49) we get

1198681= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(51)

Combining (48) and (50) we get

1198682= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(52)

Therefore from (8) (51) and (52) we have

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= sup119909119910

1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(53)

Since1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= sup0le119909le2120587

100381610038161003816100381610038161003816

119879sdot1198641

119899(119891) minus

119891 (119909)

100381610038161003816100381610038161003816

(54)

it follows from (47) and (48) that

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874((119899 + 1)

minus1

119867(

120587

119899 + 1

)) (55)

Combining (53) and (55) we get

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times[(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(56)

To prove (32) if 120596(119905) satisfies only (28) then using (34) andsecond mean value theorem for integrals we have

1198681= 119874 (120596 (120587 (119899 + 1))) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

(57)

Combining (49) and (57) we get

1198681= 119874([log (119899 + 1) 120596 ( 120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(58)

Again if 120596(119905) satisfies only (28) then using (34) and secondmean value theorem for integrals we have

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

120596 (119905)

1199052119889119905

= 119874(

1

119899 + 1

)int

119899+1

1

120596(

120587

119905

) 119889119905

= 119874(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)

(59)

Combining (50) and (59) we get

1198682= 119874([(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(60)

International Journal of Mathematics and Mathematical Sciences 7

Therefore

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times ( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(61)

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

+ 119874((

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

))

(62)

Combining (61) and (62) we get10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times (log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(63)

This completes the proof of Theorem 5

6 Applications

The theory of approximation is a very extensive field andthe study of theory of trigonometric approximation is ofgreatmathematical interest and of great practical importanceAs mentioned in [21] the 119871

119901space in general and 119871

2and

119871infin

in particular play an important role in the theory ofsignals and filters From the point of view of the applicationssharper estimates of infinite matrices [25] are useful to getbounds for the lattice norms (which occur in solid statephysics) of matrix valued functions and enable to investigateperturbations of matrix valued functions and compare themThe following corollaries may be derived fromTheorem 5

If 120596(|119909 minus 119910|) le 119861|119909 minus 119910|120572

120596lowast

(|119909 minus 119910|) le 119870|119909 minus 119910|120573

0 le

120573 lt 120572 le 1 and set

119867(119906) =

119906120572minus1

0 lt 120572 lt 1

log(1119906

) 120572 = 1

(64)

then we get Corollary 11

Corollary 11 If 119891 isin 119867120572 0 lt 120573 le 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120573

=

119874((log (119899 + 1))120573120572(119899 + 1)120573minus120572) 0 lt 120572 lt 1

119874 (log (119899 + 1) (119899 + 1)120573minus1) 120572 = 1

(65)

If we put 120573 = 0 in Corollary 11 then we get Corollary 12

Corollary 12 If 119891 isin Lip120572 0 lt 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817119888=

119874((119899 + 1)minus120572

) 0 lt 120572 lt 1

119874(

log (119899 + 1)(119899 + 1)

) 120572 = 1

(66)

7 Example

In the example we see how the sequence of averages (ie1205901

119899(119909)-means or (119862 1)mean) andNorlundmean119873

119901of partial

sums of a Fourier series is better behaved than the sequenceof partial sums 119904

119899(119909) itself

Let

119891 (119909) =

minus1 minus120587 le 119909 lt 0

1 0 le 119909 lt 120587

(67)

with 119891(119909 + 2120587) = 119891(119909) for all real 119909 Fourier series of 119891(119909) isgiven by

2

120587

infin

sum

119899=1

1 minus (minus1)119899

119899

sin 119899119909 minus120587 le 119909 le 120587 (68)

Then 119899th partial sum 119904119899(119909) of Fourier series (68) and 119899th

Cesaro sum for 120575 = 1 that is 1205901119899(119909) for the series (68) are

given by

119904119899(119909) =

4

120587

(sin119909 + 1

3

sin 3119909 + sdot sdot sdot + 1

119899

sin 119899119909)

1205901

119899(119909) =

2

120587

119899

sum

119896=1

(1 minus

119896

119899

)(

1 minus (minus1)119896

119896

) sin 119896119909(69)

FromTheorem 20 of Hardyrsquos ldquoDivergent Seriesrdquo if a Norlundmethod 119873

119901has increasing weights 119901

119899 then it is stronger

than (119862 1)Now take119873

119901to be the Norlundmatrix generated by 119901

119899=

119899 + 1 then Norlund means119873119901is given by

119905119873

119899(119891 119909) =

2

(119899 + 1) (119899 + 2)

119899

sum

119896=0

(119899 minus 119896 + 1) 119904119896(119891 119909) (70)

In this graph we observe that 1205901119899(119909) 119905119873

119899(119891 119909) converges to

119891(119909) faster than 119904119899(119909) in the interval [minus120587 120587] We further note

that near the points of discontinuities that is minus120587 0 and 120587the graph of 119904

5and 11990410

shows peaks that move closer to theline passing through points of discontinuity as n increases(Gibbs phenomenon) but in the graph of 1205901

119899(119909) 119905119873

119899(119891 119909)

8 International Journal of Mathematics and Mathematical Sciences

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(a)

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(b)

Figure 1 Graph of 119891(119909) (blue) 119904119899(119909) (pink) 1205901

119899(119909) (yellow) 119905119873

119899(119891 119909) (green) 119899 = 5 and 10

119899 = 5 10 the peaks becomeflatter (Figure 1)TheGibbs phe-nomenon is an overshoot a peculiarity of the Fourier seriesand other eigen function series at a simple discontinuitythat is the convergence of Fourier series is very slow nearthe point of discontinuity Thus the product summabilitymeans of the Fourier series of 119891(119909) overshoot the GibbsPhenomenon and show the smoothing effect of the methodThus 1205901

119899(119909) 119905119873

119899(119891 119909) is the better approximant than 119904

119899(119909) and

119873119901method is stronger than (119862 1) method

8 Conclusion

Several results concerning the degree of approximation ofperiodic signals (functions) by product summability meansof Fourier series and conjugate Fourier series in generalizedHolder metric and Holder metric have been reviewed Usinggraphical representation 120590

1

119899(119909) 119905119873

119899(119891 119909) is a better approxi-

mant to 119904119899(119909) but till now nothing has been done to show

this Some interesting application of the operator (119873119901sdot 1198641

)

used in this paper is pointed out in Remark 6 Also the resultof our theorem is more general rather than the results of anyother previous proved theorems whichwill enrich the literateof summability theory of infinite series

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude tothe anonymous learned referee(s) and the editor for theirvaluable suggestions and constructive comments whichresulted in the subsequent improvement of this paper Specialthanks are due to Professor Ricardo Estrada Academic Editorof IJMMS and Professor Mohammed Mahmoud Editorial

office Hindawi Publishing Corporation for kind cooper-ation smooth behavior during communication and theirefforts to send the reports of the paper timelyThe authors arealso grateful to all the editorial boardmembers and reviewersof Hindawi prestigious journal that is International Journalof Mathematics and Mathematical Sciences The secondauthor Kejal Khatri acknowledges the Ministry of HumanResource Development New Delhi India for providing thefinancial support and to carry out her research work underthe guidance of Dr Vishnu Narayan Mishra at SVNITSurat (Gujarat) India The first author Vishnu NarayanMishra acknowledges Cumulative Professional DevelopmentAllowance (CPDA) SVNIT Surat (Gujarat) India for sup-porting this research paper Both authors carried out theproof All the authors conceived of the study and participatedin its design and coordination Both the authors read andapproved the final manuscript

References

[1] H H Khan ldquoOn the degree of approximation of a functionsbelonging to the class Lip(120572 p)rdquo Indian Journal of Pure andApplied Mathematics vol 5 pp 132ndash136 1974

[2] H H Khan Approximation of classes of functions [PhD thesis]AMU Aligarh India 1974

[3] V N Mishra and L N Mishra ldquoTrigonometric approximationin 119871119901(119901 ge 1) spacesrdquo International Journal of Contemporary

Mathematical Sciences vol 7 pp 909ndash918 2012[4] V N Mishra K Khatri and L N Mishra ldquoProduct

(119873 119901119899)(119862 1) summability of a sequence of Fourier coefficientsrdquo

Mathematical Sciences vol 6 p 38 2012[5] VNMishra KKhatri and LNMishra ldquoProduct summability

transform of conjugate series of Fourier seriesrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2012Article ID 298923 13 pages 2012

[6] V N Mishra H H Khan K Khatri and L N Mishra ldquoOnapproximation of conjugate of signals (Functions) belongingto the generalized weighted 119882(119871

119903 120585(119905)) (119903 ge 1)-class by

product summability means of conjugate series of fourier

International Journal of Mathematics and Mathematical Sciences 9

seriesrdquo International Journal of Mathematical Analysis vol 6no 35 pp 1703ndash1715 2012

[7] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to the weighted 119882(119871

119901 120585(119905)) (119901 ge 1)-

class by almost matrix summability method of its conjugateFourier seriesrdquo International Journal of Applied Mathematicsand Mechanics vol 5 no 7 pp 16ndash27 2009

[8] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to Generalized Weighted 119882(119871

119901 120585(119905))

(119901 ge 1)-class by product summability methodrdquo Journal ofInternational Academy of Physical Sciences vol 14 no 4 pp413ndash423 2010

[9] J T Chen and Y S Jeng ldquoDual series representation and itsapplications to a string subjected to support motionsrdquoAdvancesin Engineering Software vol 27 no 3 pp 227ndash238 1996

[10] J T Chen and H-K Hong ldquoReview of dual boundary elementmethods with emphasis on hypersingular integrals and diver-gent seriesrdquo Applied Mechanics Reviews vol 52 no 1 pp 17ndash321999

[11] J T Chen H-K Hong C S Yeh and S W Chyuan ldquoInte-gral representations and regularizations for a divergent seriessolution of a beam subjected to support motionsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 9 pp 909ndash925 1996

[12] M Mursaleen and S A Mohiuddine Convergence Methods ForDouble Sequences and Applications Springer 2014

[13] G Alexits Convergence Problems of Orthogonal Series Perga-mon Elmsford New York NY USA 1961

[14] P Chandra ldquoOn the degree of approximation of a class of func-tions by means of Fourier seriesrdquo Acta Mathematica Hungaricavol 52 no 3-4 pp 199ndash205 1988

[15] P Chandra ldquoA note on the degree of approximation of contin-uous functionsrdquo Acta Mathematica Hungarica vol 62 no 1-2pp 21ndash23 1993

[16] R NMohapatra and P Chandra ldquoHolder continuous functionsand their Euler Borel and Taylor meansrdquo Math Chronicle vol11 pp 81ndash96 1982

[17] R N Mohapatra and P Chandra ldquoDegree of approximation offunctions in the Hlder metricrdquo Acta Mathematica Hungaricavol 41 no 1-2 pp 67ndash76 1983

[18] B Szal ldquoOn the rate of strong summability by matrix means inthe generalized Holder metricrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 1 article 28 2008

[19] T Singh and P Mahajan ldquoError bound of periodic signals inthe Holder metricrdquo International Journal of Mathematics andMathematical Science vol 2008 Article ID 495075 9 pages2008

[20] S Lal and K N S Yadav ldquoOn degree of approximation offunction belonging to the Lipschitz class by (119862 1)(119864 1)meansof its Fourier seriesrdquo Bulletin of Calcutta Mathematical Societyvol 93 pp 191ndash196 2001

[21] E Z Psarakis and G V Moustakides ldquoAn L2-based method

for the design of 1-D zero phase FIR digital filtersrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 44 no 7 pp 591ndash601 1997

[22] G Das T Ghosh and B K Ray ldquoDegree of approximation offunctions in the Holder metric by (e c) Meansrdquo Proceedings ofthe Indian Academy of Science vol 105 pp 315ndash327 1995

[23] A ZygmundTrigonometric Series CambridgeUniversity PressLondon UK 3rd edition 2002

[24] G Bachman L Narici and E Beckenstien Fourier andWaveletAnalysis Springer New York NY USA 2000

[25] M I Gilrsquo ldquoEstimates for entries of matrix valued functions ofinfinite matricesrdquoMathematical Physics Analysis and Geometryvol 11 no 2 pp 175ndash186 2008

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 International Journal of Mathematics and Mathematical Sciences

Now using (33) Lemma 8 we get

1198681= 119874(

1

119899 + 1

) int

120587(119899+1)

0

10038161003816100381610038161003816120595119909(119905) minus 120595

119910(119905)

10038161003816100381610038161003816

119905

119889119905

= 119874(

1

119899 + 1

) int

120587(119899+1)

0

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

119905

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874 (log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(49)

Now using (33) Lemma 9 we get

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

119905minus2

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816) 119889119905

= 119874(

1

119899 + 1

120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))int

120587

120587(119899+1)

119905minus2

119889119905

= 119874 (120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816))

(50)

Using the fact that we may write 119868119896= 1198681minus120573120572

119896119868120573120572

119896 119896 = 1 2

and combining (47) and (49) we get

1198681= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(51)

Combining (48) and (50) we get

1198682= 119874([(119899 + 1)

minus1

119867(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(52)

Therefore from (8) (51) and (52) we have

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= sup119909119910

1003816100381610038161003816119864119899(119909) minus 119864

119899(119910)

1003816100381610038161003816

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times [(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(53)

Since1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= sup0le119909le2120587

100381610038161003816100381610038161003816

119879sdot1198641

119899(119891) minus

119891 (119909)

100381610038161003816100381610038161003816

(54)

it follows from (47) and (48) that

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874((119899 + 1)

minus1

119867(

120587

119899 + 1

)) (55)

Combining (53) and (55) we get

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

(log (119899 + 1))120573120572

times[(119899 + 1)minus1

119867(

120587

119899 + 1

)]

1minus120573120572

(56)

To prove (32) if 120596(119905) satisfies only (28) then using (34) andsecond mean value theorem for integrals we have

1198681= 119874 (120596 (120587 (119899 + 1))) int

120587(119899+1)

0

1

119905

119889119905

where 0 lt 119905 le 120587

(119899 + 1)

= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

(57)

Combining (49) and (57) we get

1198681= 119874([log (119899 + 1) 120596 ( 120587

119899 + 1

)]

1minus120573120572

times [log (119899 + 1) 120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(58)

Again if 120596(119905) satisfies only (28) then using (34) and secondmean value theorem for integrals we have

1198682= 119874(

1

119899 + 1

)int

120587

120587(119899+1)

120596 (119905)

1199052119889119905

= 119874(

1

119899 + 1

)int

119899+1

1

120596(

120587

119905

) 119889119905

= 119874(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)

(59)

Combining (50) and (59) we get

1198682= 119874([(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

[120596 (1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)]120573120572

)

(60)

International Journal of Mathematics and Mathematical Sciences 7

Therefore

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times ( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(61)

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

+ 119874((

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

))

(62)

Combining (61) and (62) we get10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times (log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(63)

This completes the proof of Theorem 5

6 Applications

The theory of approximation is a very extensive field andthe study of theory of trigonometric approximation is ofgreatmathematical interest and of great practical importanceAs mentioned in [21] the 119871

119901space in general and 119871

2and

119871infin

in particular play an important role in the theory ofsignals and filters From the point of view of the applicationssharper estimates of infinite matrices [25] are useful to getbounds for the lattice norms (which occur in solid statephysics) of matrix valued functions and enable to investigateperturbations of matrix valued functions and compare themThe following corollaries may be derived fromTheorem 5

If 120596(|119909 minus 119910|) le 119861|119909 minus 119910|120572

120596lowast

(|119909 minus 119910|) le 119870|119909 minus 119910|120573

0 le

120573 lt 120572 le 1 and set

119867(119906) =

119906120572minus1

0 lt 120572 lt 1

log(1119906

) 120572 = 1

(64)

then we get Corollary 11

Corollary 11 If 119891 isin 119867120572 0 lt 120573 le 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120573

=

119874((log (119899 + 1))120573120572(119899 + 1)120573minus120572) 0 lt 120572 lt 1

119874 (log (119899 + 1) (119899 + 1)120573minus1) 120572 = 1

(65)

If we put 120573 = 0 in Corollary 11 then we get Corollary 12

Corollary 12 If 119891 isin Lip120572 0 lt 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817119888=

119874((119899 + 1)minus120572

) 0 lt 120572 lt 1

119874(

log (119899 + 1)(119899 + 1)

) 120572 = 1

(66)

7 Example

In the example we see how the sequence of averages (ie1205901

119899(119909)-means or (119862 1)mean) andNorlundmean119873

119901of partial

sums of a Fourier series is better behaved than the sequenceof partial sums 119904

119899(119909) itself

Let

119891 (119909) =

minus1 minus120587 le 119909 lt 0

1 0 le 119909 lt 120587

(67)

with 119891(119909 + 2120587) = 119891(119909) for all real 119909 Fourier series of 119891(119909) isgiven by

2

120587

infin

sum

119899=1

1 minus (minus1)119899

119899

sin 119899119909 minus120587 le 119909 le 120587 (68)

Then 119899th partial sum 119904119899(119909) of Fourier series (68) and 119899th

Cesaro sum for 120575 = 1 that is 1205901119899(119909) for the series (68) are

given by

119904119899(119909) =

4

120587

(sin119909 + 1

3

sin 3119909 + sdot sdot sdot + 1

119899

sin 119899119909)

1205901

119899(119909) =

2

120587

119899

sum

119896=1

(1 minus

119896

119899

)(

1 minus (minus1)119896

119896

) sin 119896119909(69)

FromTheorem 20 of Hardyrsquos ldquoDivergent Seriesrdquo if a Norlundmethod 119873

119901has increasing weights 119901

119899 then it is stronger

than (119862 1)Now take119873

119901to be the Norlundmatrix generated by 119901

119899=

119899 + 1 then Norlund means119873119901is given by

119905119873

119899(119891 119909) =

2

(119899 + 1) (119899 + 2)

119899

sum

119896=0

(119899 minus 119896 + 1) 119904119896(119891 119909) (70)

In this graph we observe that 1205901119899(119909) 119905119873

119899(119891 119909) converges to

119891(119909) faster than 119904119899(119909) in the interval [minus120587 120587] We further note

that near the points of discontinuities that is minus120587 0 and 120587the graph of 119904

5and 11990410

shows peaks that move closer to theline passing through points of discontinuity as n increases(Gibbs phenomenon) but in the graph of 1205901

119899(119909) 119905119873

119899(119891 119909)

8 International Journal of Mathematics and Mathematical Sciences

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(a)

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(b)

Figure 1 Graph of 119891(119909) (blue) 119904119899(119909) (pink) 1205901

119899(119909) (yellow) 119905119873

119899(119891 119909) (green) 119899 = 5 and 10

119899 = 5 10 the peaks becomeflatter (Figure 1)TheGibbs phe-nomenon is an overshoot a peculiarity of the Fourier seriesand other eigen function series at a simple discontinuitythat is the convergence of Fourier series is very slow nearthe point of discontinuity Thus the product summabilitymeans of the Fourier series of 119891(119909) overshoot the GibbsPhenomenon and show the smoothing effect of the methodThus 1205901

119899(119909) 119905119873

119899(119891 119909) is the better approximant than 119904

119899(119909) and

119873119901method is stronger than (119862 1) method

8 Conclusion

Several results concerning the degree of approximation ofperiodic signals (functions) by product summability meansof Fourier series and conjugate Fourier series in generalizedHolder metric and Holder metric have been reviewed Usinggraphical representation 120590

1

119899(119909) 119905119873

119899(119891 119909) is a better approxi-

mant to 119904119899(119909) but till now nothing has been done to show

this Some interesting application of the operator (119873119901sdot 1198641

)

used in this paper is pointed out in Remark 6 Also the resultof our theorem is more general rather than the results of anyother previous proved theorems whichwill enrich the literateof summability theory of infinite series

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude tothe anonymous learned referee(s) and the editor for theirvaluable suggestions and constructive comments whichresulted in the subsequent improvement of this paper Specialthanks are due to Professor Ricardo Estrada Academic Editorof IJMMS and Professor Mohammed Mahmoud Editorial

office Hindawi Publishing Corporation for kind cooper-ation smooth behavior during communication and theirefforts to send the reports of the paper timelyThe authors arealso grateful to all the editorial boardmembers and reviewersof Hindawi prestigious journal that is International Journalof Mathematics and Mathematical Sciences The secondauthor Kejal Khatri acknowledges the Ministry of HumanResource Development New Delhi India for providing thefinancial support and to carry out her research work underthe guidance of Dr Vishnu Narayan Mishra at SVNITSurat (Gujarat) India The first author Vishnu NarayanMishra acknowledges Cumulative Professional DevelopmentAllowance (CPDA) SVNIT Surat (Gujarat) India for sup-porting this research paper Both authors carried out theproof All the authors conceived of the study and participatedin its design and coordination Both the authors read andapproved the final manuscript

References

[1] H H Khan ldquoOn the degree of approximation of a functionsbelonging to the class Lip(120572 p)rdquo Indian Journal of Pure andApplied Mathematics vol 5 pp 132ndash136 1974

[2] H H Khan Approximation of classes of functions [PhD thesis]AMU Aligarh India 1974

[3] V N Mishra and L N Mishra ldquoTrigonometric approximationin 119871119901(119901 ge 1) spacesrdquo International Journal of Contemporary

Mathematical Sciences vol 7 pp 909ndash918 2012[4] V N Mishra K Khatri and L N Mishra ldquoProduct

(119873 119901119899)(119862 1) summability of a sequence of Fourier coefficientsrdquo

Mathematical Sciences vol 6 p 38 2012[5] VNMishra KKhatri and LNMishra ldquoProduct summability

transform of conjugate series of Fourier seriesrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2012Article ID 298923 13 pages 2012

[6] V N Mishra H H Khan K Khatri and L N Mishra ldquoOnapproximation of conjugate of signals (Functions) belongingto the generalized weighted 119882(119871

119903 120585(119905)) (119903 ge 1)-class by

product summability means of conjugate series of fourier

International Journal of Mathematics and Mathematical Sciences 9

seriesrdquo International Journal of Mathematical Analysis vol 6no 35 pp 1703ndash1715 2012

[7] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to the weighted 119882(119871

119901 120585(119905)) (119901 ge 1)-

class by almost matrix summability method of its conjugateFourier seriesrdquo International Journal of Applied Mathematicsand Mechanics vol 5 no 7 pp 16ndash27 2009

[8] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to Generalized Weighted 119882(119871

119901 120585(119905))

(119901 ge 1)-class by product summability methodrdquo Journal ofInternational Academy of Physical Sciences vol 14 no 4 pp413ndash423 2010

[9] J T Chen and Y S Jeng ldquoDual series representation and itsapplications to a string subjected to support motionsrdquoAdvancesin Engineering Software vol 27 no 3 pp 227ndash238 1996

[10] J T Chen and H-K Hong ldquoReview of dual boundary elementmethods with emphasis on hypersingular integrals and diver-gent seriesrdquo Applied Mechanics Reviews vol 52 no 1 pp 17ndash321999

[11] J T Chen H-K Hong C S Yeh and S W Chyuan ldquoInte-gral representations and regularizations for a divergent seriessolution of a beam subjected to support motionsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 9 pp 909ndash925 1996

[12] M Mursaleen and S A Mohiuddine Convergence Methods ForDouble Sequences and Applications Springer 2014

[13] G Alexits Convergence Problems of Orthogonal Series Perga-mon Elmsford New York NY USA 1961

[14] P Chandra ldquoOn the degree of approximation of a class of func-tions by means of Fourier seriesrdquo Acta Mathematica Hungaricavol 52 no 3-4 pp 199ndash205 1988

[15] P Chandra ldquoA note on the degree of approximation of contin-uous functionsrdquo Acta Mathematica Hungarica vol 62 no 1-2pp 21ndash23 1993

[16] R NMohapatra and P Chandra ldquoHolder continuous functionsand their Euler Borel and Taylor meansrdquo Math Chronicle vol11 pp 81ndash96 1982

[17] R N Mohapatra and P Chandra ldquoDegree of approximation offunctions in the Hlder metricrdquo Acta Mathematica Hungaricavol 41 no 1-2 pp 67ndash76 1983

[18] B Szal ldquoOn the rate of strong summability by matrix means inthe generalized Holder metricrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 1 article 28 2008

[19] T Singh and P Mahajan ldquoError bound of periodic signals inthe Holder metricrdquo International Journal of Mathematics andMathematical Science vol 2008 Article ID 495075 9 pages2008

[20] S Lal and K N S Yadav ldquoOn degree of approximation offunction belonging to the Lipschitz class by (119862 1)(119864 1)meansof its Fourier seriesrdquo Bulletin of Calcutta Mathematical Societyvol 93 pp 191ndash196 2001

[21] E Z Psarakis and G V Moustakides ldquoAn L2-based method

for the design of 1-D zero phase FIR digital filtersrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 44 no 7 pp 591ndash601 1997

[22] G Das T Ghosh and B K Ray ldquoDegree of approximation offunctions in the Holder metric by (e c) Meansrdquo Proceedings ofthe Indian Academy of Science vol 105 pp 315ndash327 1995

[23] A ZygmundTrigonometric Series CambridgeUniversity PressLondon UK 3rd edition 2002

[24] G Bachman L Narici and E Beckenstien Fourier andWaveletAnalysis Springer New York NY USA 2000

[25] M I Gilrsquo ldquoEstimates for entries of matrix valued functions ofinfinite matricesrdquoMathematical Physics Analysis and Geometryvol 11 no 2 pp 175ndash186 2008

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

International Journal of Mathematics and Mathematical Sciences 7

Therefore

sup119909119910

100381610038161003816100381610038161003816

Δ120596lowast

119864 (119909 119910)

100381610038161003816100381610038161003816

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times ( log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(61)

1003817100381710038171003817119864119899(119909)

1003817100381710038171003817119888= 119874(log (119899 + 1) 120596 ( 120587

119899 + 1

))

+ 119874((

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

))

(62)

Combining (61) and (62) we get10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120596lowast

= 119874

120596(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)120573120572

120596lowast(1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)

times (log (119899 + 1) [120596 ( 120587

119899 + 1

)]

1minus120573120572

+[(

1

119899 + 1

)

119899

sum

119896=0

120596(

120587

119899 + 1

)]

1minus120573120572

)

(63)

This completes the proof of Theorem 5

6 Applications

The theory of approximation is a very extensive field andthe study of theory of trigonometric approximation is ofgreatmathematical interest and of great practical importanceAs mentioned in [21] the 119871

119901space in general and 119871

2and

119871infin

in particular play an important role in the theory ofsignals and filters From the point of view of the applicationssharper estimates of infinite matrices [25] are useful to getbounds for the lattice norms (which occur in solid statephysics) of matrix valued functions and enable to investigateperturbations of matrix valued functions and compare themThe following corollaries may be derived fromTheorem 5

If 120596(|119909 minus 119910|) le 119861|119909 minus 119910|120572

120596lowast

(|119909 minus 119910|) le 119870|119909 minus 119910|120573

0 le

120573 lt 120572 le 1 and set

119867(119906) =

119906120572minus1

0 lt 120572 lt 1

log(1119906

) 120572 = 1

(64)

then we get Corollary 11

Corollary 11 If 119891 isin 119867120572 0 lt 120573 le 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817120573

=

119874((log (119899 + 1))120573120572(119899 + 1)120573minus120572) 0 lt 120572 lt 1

119874 (log (119899 + 1) (119899 + 1)120573minus1) 120572 = 1

(65)

If we put 120573 = 0 in Corollary 11 then we get Corollary 12

Corollary 12 If 119891 isin Lip120572 0 lt 120572 le 1 then

10038171003817100381710038171003817119873119864

119899(119891) minus

119891 (119909)

10038171003817100381710038171003817119888=

119874((119899 + 1)minus120572

) 0 lt 120572 lt 1

119874(

log (119899 + 1)(119899 + 1)

) 120572 = 1

(66)

7 Example

In the example we see how the sequence of averages (ie1205901

119899(119909)-means or (119862 1)mean) andNorlundmean119873

119901of partial

sums of a Fourier series is better behaved than the sequenceof partial sums 119904

119899(119909) itself

Let

119891 (119909) =

minus1 minus120587 le 119909 lt 0

1 0 le 119909 lt 120587

(67)

with 119891(119909 + 2120587) = 119891(119909) for all real 119909 Fourier series of 119891(119909) isgiven by

2

120587

infin

sum

119899=1

1 minus (minus1)119899

119899

sin 119899119909 minus120587 le 119909 le 120587 (68)

Then 119899th partial sum 119904119899(119909) of Fourier series (68) and 119899th

Cesaro sum for 120575 = 1 that is 1205901119899(119909) for the series (68) are

given by

119904119899(119909) =

4

120587

(sin119909 + 1

3

sin 3119909 + sdot sdot sdot + 1

119899

sin 119899119909)

1205901

119899(119909) =

2

120587

119899

sum

119896=1

(1 minus

119896

119899

)(

1 minus (minus1)119896

119896

) sin 119896119909(69)

FromTheorem 20 of Hardyrsquos ldquoDivergent Seriesrdquo if a Norlundmethod 119873

119901has increasing weights 119901

119899 then it is stronger

than (119862 1)Now take119873

119901to be the Norlundmatrix generated by 119901

119899=

119899 + 1 then Norlund means119873119901is given by

119905119873

119899(119891 119909) =

2

(119899 + 1) (119899 + 2)

119899

sum

119896=0

(119899 minus 119896 + 1) 119904119896(119891 119909) (70)

In this graph we observe that 1205901119899(119909) 119905119873

119899(119891 119909) converges to

119891(119909) faster than 119904119899(119909) in the interval [minus120587 120587] We further note

that near the points of discontinuities that is minus120587 0 and 120587the graph of 119904

5and 11990410

shows peaks that move closer to theline passing through points of discontinuity as n increases(Gibbs phenomenon) but in the graph of 1205901

119899(119909) 119905119873

119899(119891 119909)

8 International Journal of Mathematics and Mathematical Sciences

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(a)

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(b)

Figure 1 Graph of 119891(119909) (blue) 119904119899(119909) (pink) 1205901

119899(119909) (yellow) 119905119873

119899(119891 119909) (green) 119899 = 5 and 10

119899 = 5 10 the peaks becomeflatter (Figure 1)TheGibbs phe-nomenon is an overshoot a peculiarity of the Fourier seriesand other eigen function series at a simple discontinuitythat is the convergence of Fourier series is very slow nearthe point of discontinuity Thus the product summabilitymeans of the Fourier series of 119891(119909) overshoot the GibbsPhenomenon and show the smoothing effect of the methodThus 1205901

119899(119909) 119905119873

119899(119891 119909) is the better approximant than 119904

119899(119909) and

119873119901method is stronger than (119862 1) method

8 Conclusion

Several results concerning the degree of approximation ofperiodic signals (functions) by product summability meansof Fourier series and conjugate Fourier series in generalizedHolder metric and Holder metric have been reviewed Usinggraphical representation 120590

1

119899(119909) 119905119873

119899(119891 119909) is a better approxi-

mant to 119904119899(119909) but till now nothing has been done to show

this Some interesting application of the operator (119873119901sdot 1198641

)

used in this paper is pointed out in Remark 6 Also the resultof our theorem is more general rather than the results of anyother previous proved theorems whichwill enrich the literateof summability theory of infinite series

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude tothe anonymous learned referee(s) and the editor for theirvaluable suggestions and constructive comments whichresulted in the subsequent improvement of this paper Specialthanks are due to Professor Ricardo Estrada Academic Editorof IJMMS and Professor Mohammed Mahmoud Editorial

office Hindawi Publishing Corporation for kind cooper-ation smooth behavior during communication and theirefforts to send the reports of the paper timelyThe authors arealso grateful to all the editorial boardmembers and reviewersof Hindawi prestigious journal that is International Journalof Mathematics and Mathematical Sciences The secondauthor Kejal Khatri acknowledges the Ministry of HumanResource Development New Delhi India for providing thefinancial support and to carry out her research work underthe guidance of Dr Vishnu Narayan Mishra at SVNITSurat (Gujarat) India The first author Vishnu NarayanMishra acknowledges Cumulative Professional DevelopmentAllowance (CPDA) SVNIT Surat (Gujarat) India for sup-porting this research paper Both authors carried out theproof All the authors conceived of the study and participatedin its design and coordination Both the authors read andapproved the final manuscript

References

[1] H H Khan ldquoOn the degree of approximation of a functionsbelonging to the class Lip(120572 p)rdquo Indian Journal of Pure andApplied Mathematics vol 5 pp 132ndash136 1974

[2] H H Khan Approximation of classes of functions [PhD thesis]AMU Aligarh India 1974

[3] V N Mishra and L N Mishra ldquoTrigonometric approximationin 119871119901(119901 ge 1) spacesrdquo International Journal of Contemporary

Mathematical Sciences vol 7 pp 909ndash918 2012[4] V N Mishra K Khatri and L N Mishra ldquoProduct

(119873 119901119899)(119862 1) summability of a sequence of Fourier coefficientsrdquo

Mathematical Sciences vol 6 p 38 2012[5] VNMishra KKhatri and LNMishra ldquoProduct summability

transform of conjugate series of Fourier seriesrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2012Article ID 298923 13 pages 2012

[6] V N Mishra H H Khan K Khatri and L N Mishra ldquoOnapproximation of conjugate of signals (Functions) belongingto the generalized weighted 119882(119871

119903 120585(119905)) (119903 ge 1)-class by

product summability means of conjugate series of fourier

International Journal of Mathematics and Mathematical Sciences 9

seriesrdquo International Journal of Mathematical Analysis vol 6no 35 pp 1703ndash1715 2012

[7] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to the weighted 119882(119871

119901 120585(119905)) (119901 ge 1)-

class by almost matrix summability method of its conjugateFourier seriesrdquo International Journal of Applied Mathematicsand Mechanics vol 5 no 7 pp 16ndash27 2009

[8] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to Generalized Weighted 119882(119871

119901 120585(119905))

(119901 ge 1)-class by product summability methodrdquo Journal ofInternational Academy of Physical Sciences vol 14 no 4 pp413ndash423 2010

[9] J T Chen and Y S Jeng ldquoDual series representation and itsapplications to a string subjected to support motionsrdquoAdvancesin Engineering Software vol 27 no 3 pp 227ndash238 1996

[10] J T Chen and H-K Hong ldquoReview of dual boundary elementmethods with emphasis on hypersingular integrals and diver-gent seriesrdquo Applied Mechanics Reviews vol 52 no 1 pp 17ndash321999

[11] J T Chen H-K Hong C S Yeh and S W Chyuan ldquoInte-gral representations and regularizations for a divergent seriessolution of a beam subjected to support motionsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 9 pp 909ndash925 1996

[12] M Mursaleen and S A Mohiuddine Convergence Methods ForDouble Sequences and Applications Springer 2014

[13] G Alexits Convergence Problems of Orthogonal Series Perga-mon Elmsford New York NY USA 1961

[14] P Chandra ldquoOn the degree of approximation of a class of func-tions by means of Fourier seriesrdquo Acta Mathematica Hungaricavol 52 no 3-4 pp 199ndash205 1988

[15] P Chandra ldquoA note on the degree of approximation of contin-uous functionsrdquo Acta Mathematica Hungarica vol 62 no 1-2pp 21ndash23 1993

[16] R NMohapatra and P Chandra ldquoHolder continuous functionsand their Euler Borel and Taylor meansrdquo Math Chronicle vol11 pp 81ndash96 1982

[17] R N Mohapatra and P Chandra ldquoDegree of approximation offunctions in the Hlder metricrdquo Acta Mathematica Hungaricavol 41 no 1-2 pp 67ndash76 1983

[18] B Szal ldquoOn the rate of strong summability by matrix means inthe generalized Holder metricrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 1 article 28 2008

[19] T Singh and P Mahajan ldquoError bound of periodic signals inthe Holder metricrdquo International Journal of Mathematics andMathematical Science vol 2008 Article ID 495075 9 pages2008

[20] S Lal and K N S Yadav ldquoOn degree of approximation offunction belonging to the Lipschitz class by (119862 1)(119864 1)meansof its Fourier seriesrdquo Bulletin of Calcutta Mathematical Societyvol 93 pp 191ndash196 2001

[21] E Z Psarakis and G V Moustakides ldquoAn L2-based method

for the design of 1-D zero phase FIR digital filtersrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 44 no 7 pp 591ndash601 1997

[22] G Das T Ghosh and B K Ray ldquoDegree of approximation offunctions in the Holder metric by (e c) Meansrdquo Proceedings ofthe Indian Academy of Science vol 105 pp 315ndash327 1995

[23] A ZygmundTrigonometric Series CambridgeUniversity PressLondon UK 3rd edition 2002

[24] G Bachman L Narici and E Beckenstien Fourier andWaveletAnalysis Springer New York NY USA 2000

[25] M I Gilrsquo ldquoEstimates for entries of matrix valued functions ofinfinite matricesrdquoMathematical Physics Analysis and Geometryvol 11 no 2 pp 175ndash186 2008

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 International Journal of Mathematics and Mathematical Sciences

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(a)

minus3 minus2 minus1 1 2 3

10

05

05

10

x

y

(b)

Figure 1 Graph of 119891(119909) (blue) 119904119899(119909) (pink) 1205901

119899(119909) (yellow) 119905119873

119899(119891 119909) (green) 119899 = 5 and 10

119899 = 5 10 the peaks becomeflatter (Figure 1)TheGibbs phe-nomenon is an overshoot a peculiarity of the Fourier seriesand other eigen function series at a simple discontinuitythat is the convergence of Fourier series is very slow nearthe point of discontinuity Thus the product summabilitymeans of the Fourier series of 119891(119909) overshoot the GibbsPhenomenon and show the smoothing effect of the methodThus 1205901

119899(119909) 119905119873

119899(119891 119909) is the better approximant than 119904

119899(119909) and

119873119901method is stronger than (119862 1) method

8 Conclusion

Several results concerning the degree of approximation ofperiodic signals (functions) by product summability meansof Fourier series and conjugate Fourier series in generalizedHolder metric and Holder metric have been reviewed Usinggraphical representation 120590

1

119899(119909) 119905119873

119899(119891 119909) is a better approxi-

mant to 119904119899(119909) but till now nothing has been done to show

this Some interesting application of the operator (119873119901sdot 1198641

)

used in this paper is pointed out in Remark 6 Also the resultof our theorem is more general rather than the results of anyother previous proved theorems whichwill enrich the literateof summability theory of infinite series

Conflict of Interests

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

Acknowledgments

The authors would like to express their deep gratitude tothe anonymous learned referee(s) and the editor for theirvaluable suggestions and constructive comments whichresulted in the subsequent improvement of this paper Specialthanks are due to Professor Ricardo Estrada Academic Editorof IJMMS and Professor Mohammed Mahmoud Editorial

office Hindawi Publishing Corporation for kind cooper-ation smooth behavior during communication and theirefforts to send the reports of the paper timelyThe authors arealso grateful to all the editorial boardmembers and reviewersof Hindawi prestigious journal that is International Journalof Mathematics and Mathematical Sciences The secondauthor Kejal Khatri acknowledges the Ministry of HumanResource Development New Delhi India for providing thefinancial support and to carry out her research work underthe guidance of Dr Vishnu Narayan Mishra at SVNITSurat (Gujarat) India The first author Vishnu NarayanMishra acknowledges Cumulative Professional DevelopmentAllowance (CPDA) SVNIT Surat (Gujarat) India for sup-porting this research paper Both authors carried out theproof All the authors conceived of the study and participatedin its design and coordination Both the authors read andapproved the final manuscript

References

[1] H H Khan ldquoOn the degree of approximation of a functionsbelonging to the class Lip(120572 p)rdquo Indian Journal of Pure andApplied Mathematics vol 5 pp 132ndash136 1974

[2] H H Khan Approximation of classes of functions [PhD thesis]AMU Aligarh India 1974

[3] V N Mishra and L N Mishra ldquoTrigonometric approximationin 119871119901(119901 ge 1) spacesrdquo International Journal of Contemporary

Mathematical Sciences vol 7 pp 909ndash918 2012[4] V N Mishra K Khatri and L N Mishra ldquoProduct

(119873 119901119899)(119862 1) summability of a sequence of Fourier coefficientsrdquo

Mathematical Sciences vol 6 p 38 2012[5] VNMishra KKhatri and LNMishra ldquoProduct summability

transform of conjugate series of Fourier seriesrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2012Article ID 298923 13 pages 2012

[6] V N Mishra H H Khan K Khatri and L N Mishra ldquoOnapproximation of conjugate of signals (Functions) belongingto the generalized weighted 119882(119871

119903 120585(119905)) (119903 ge 1)-class by

product summability means of conjugate series of fourier

International Journal of Mathematics and Mathematical Sciences 9

seriesrdquo International Journal of Mathematical Analysis vol 6no 35 pp 1703ndash1715 2012

[7] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to the weighted 119882(119871

119901 120585(119905)) (119901 ge 1)-

class by almost matrix summability method of its conjugateFourier seriesrdquo International Journal of Applied Mathematicsand Mechanics vol 5 no 7 pp 16ndash27 2009

[8] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to Generalized Weighted 119882(119871

119901 120585(119905))

(119901 ge 1)-class by product summability methodrdquo Journal ofInternational Academy of Physical Sciences vol 14 no 4 pp413ndash423 2010

[9] J T Chen and Y S Jeng ldquoDual series representation and itsapplications to a string subjected to support motionsrdquoAdvancesin Engineering Software vol 27 no 3 pp 227ndash238 1996

[10] J T Chen and H-K Hong ldquoReview of dual boundary elementmethods with emphasis on hypersingular integrals and diver-gent seriesrdquo Applied Mechanics Reviews vol 52 no 1 pp 17ndash321999

[11] J T Chen H-K Hong C S Yeh and S W Chyuan ldquoInte-gral representations and regularizations for a divergent seriessolution of a beam subjected to support motionsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 9 pp 909ndash925 1996

[12] M Mursaleen and S A Mohiuddine Convergence Methods ForDouble Sequences and Applications Springer 2014

[13] G Alexits Convergence Problems of Orthogonal Series Perga-mon Elmsford New York NY USA 1961

[14] P Chandra ldquoOn the degree of approximation of a class of func-tions by means of Fourier seriesrdquo Acta Mathematica Hungaricavol 52 no 3-4 pp 199ndash205 1988

[15] P Chandra ldquoA note on the degree of approximation of contin-uous functionsrdquo Acta Mathematica Hungarica vol 62 no 1-2pp 21ndash23 1993

[16] R NMohapatra and P Chandra ldquoHolder continuous functionsand their Euler Borel and Taylor meansrdquo Math Chronicle vol11 pp 81ndash96 1982

[17] R N Mohapatra and P Chandra ldquoDegree of approximation offunctions in the Hlder metricrdquo Acta Mathematica Hungaricavol 41 no 1-2 pp 67ndash76 1983

[18] B Szal ldquoOn the rate of strong summability by matrix means inthe generalized Holder metricrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 1 article 28 2008

[19] T Singh and P Mahajan ldquoError bound of periodic signals inthe Holder metricrdquo International Journal of Mathematics andMathematical Science vol 2008 Article ID 495075 9 pages2008

[20] S Lal and K N S Yadav ldquoOn degree of approximation offunction belonging to the Lipschitz class by (119862 1)(119864 1)meansof its Fourier seriesrdquo Bulletin of Calcutta Mathematical Societyvol 93 pp 191ndash196 2001

[21] E Z Psarakis and G V Moustakides ldquoAn L2-based method

for the design of 1-D zero phase FIR digital filtersrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 44 no 7 pp 591ndash601 1997

[22] G Das T Ghosh and B K Ray ldquoDegree of approximation offunctions in the Holder metric by (e c) Meansrdquo Proceedings ofthe Indian Academy of Science vol 105 pp 315ndash327 1995

[23] A ZygmundTrigonometric Series CambridgeUniversity PressLondon UK 3rd edition 2002

[24] G Bachman L Narici and E Beckenstien Fourier andWaveletAnalysis Springer New York NY USA 2000

[25] M I Gilrsquo ldquoEstimates for entries of matrix valued functions ofinfinite matricesrdquoMathematical Physics Analysis and Geometryvol 11 no 2 pp 175ndash186 2008

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

International Journal of Mathematics and Mathematical Sciences 9

seriesrdquo International Journal of Mathematical Analysis vol 6no 35 pp 1703ndash1715 2012

[7] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to the weighted 119882(119871

119901 120585(119905)) (119901 ge 1)-

class by almost matrix summability method of its conjugateFourier seriesrdquo International Journal of Applied Mathematicsand Mechanics vol 5 no 7 pp 16ndash27 2009

[8] V N Mishra ldquoOn the degree of approximation of signals(Functions) belonging to Generalized Weighted 119882(119871

119901 120585(119905))

(119901 ge 1)-class by product summability methodrdquo Journal ofInternational Academy of Physical Sciences vol 14 no 4 pp413ndash423 2010

[9] J T Chen and Y S Jeng ldquoDual series representation and itsapplications to a string subjected to support motionsrdquoAdvancesin Engineering Software vol 27 no 3 pp 227ndash238 1996

[10] J T Chen and H-K Hong ldquoReview of dual boundary elementmethods with emphasis on hypersingular integrals and diver-gent seriesrdquo Applied Mechanics Reviews vol 52 no 1 pp 17ndash321999

[11] J T Chen H-K Hong C S Yeh and S W Chyuan ldquoInte-gral representations and regularizations for a divergent seriessolution of a beam subjected to support motionsrdquo EarthquakeEngineering and Structural Dynamics vol 25 no 9 pp 909ndash925 1996

[12] M Mursaleen and S A Mohiuddine Convergence Methods ForDouble Sequences and Applications Springer 2014

[13] G Alexits Convergence Problems of Orthogonal Series Perga-mon Elmsford New York NY USA 1961

[14] P Chandra ldquoOn the degree of approximation of a class of func-tions by means of Fourier seriesrdquo Acta Mathematica Hungaricavol 52 no 3-4 pp 199ndash205 1988

[15] P Chandra ldquoA note on the degree of approximation of contin-uous functionsrdquo Acta Mathematica Hungarica vol 62 no 1-2pp 21ndash23 1993

[16] R NMohapatra and P Chandra ldquoHolder continuous functionsand their Euler Borel and Taylor meansrdquo Math Chronicle vol11 pp 81ndash96 1982

[17] R N Mohapatra and P Chandra ldquoDegree of approximation offunctions in the Hlder metricrdquo Acta Mathematica Hungaricavol 41 no 1-2 pp 67ndash76 1983

[18] B Szal ldquoOn the rate of strong summability by matrix means inthe generalized Holder metricrdquo Journal of Inequalities in Pureand Applied Mathematics vol 9 no 1 article 28 2008

[19] T Singh and P Mahajan ldquoError bound of periodic signals inthe Holder metricrdquo International Journal of Mathematics andMathematical Science vol 2008 Article ID 495075 9 pages2008

[20] S Lal and K N S Yadav ldquoOn degree of approximation offunction belonging to the Lipschitz class by (119862 1)(119864 1)meansof its Fourier seriesrdquo Bulletin of Calcutta Mathematical Societyvol 93 pp 191ndash196 2001

[21] E Z Psarakis and G V Moustakides ldquoAn L2-based method

for the design of 1-D zero phase FIR digital filtersrdquo IEEETransactions on Circuits and Systems I FundamentalTheory andApplications vol 44 no 7 pp 591ndash601 1997

[22] G Das T Ghosh and B K Ray ldquoDegree of approximation offunctions in the Holder metric by (e c) Meansrdquo Proceedings ofthe Indian Academy of Science vol 105 pp 315ndash327 1995

[23] A ZygmundTrigonometric Series CambridgeUniversity PressLondon UK 3rd edition 2002

[24] G Bachman L Narici and E Beckenstien Fourier andWaveletAnalysis Springer New York NY USA 2000

[25] M I Gilrsquo ldquoEstimates for entries of matrix valued functions ofinfinite matricesrdquoMathematical Physics Analysis and Geometryvol 11 no 2 pp 175ndash186 2008

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

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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