American Journal of Applied Mathematics and Statistics, 2014, Vol. 2, No. 5, 352-356 Available online at http://pubs.sciepub.com/ajams/2/5/9 © Science and Education Publishing DOI:10.12691/ajams-2-5-9

Approximation of Conjugate Series of the Fourier Series of a Function of Class W(Lp,ξ(t)) by Product Means

M. Misra1, B. Majhi2, B.P. Padhy3, P. Samanta4, U.K. Misra5,*

1Department of Mathematics Binayak Acharya College, Berhampur, Odisha, India 2Department of Mathematics GIET, Gunupur, Odisha, India

3Department of Mathematics Roland Institute of Technology, Golanthara, Odisha, India 4Department of Mathematics Berhampur University, Berhampur, Odisha, India

5Department of Mathematics National Institute of Science and Technology Pallur Hills, Golanthara, Odisha, India *Corresponding author: umakanta_misra@yahoo.com

Received September 27, 2014; Revised October 20, 2014; Accepted October 29, 2014

Abstract In this paper a theorem on degree of approximation of a function ( )( ),pf W L tξ∈ by product

summability ( )( ), , nE q N p of the conjugate series of Fourier series associated with f has been established.

Keywords: degree of approximation, ( )( ),pW L tξ class of function, ( ),E q mean, ( ), nN p mean, ( )( ), , nE q N p

product mean, Fourier series, conjugate series, Lebesgue integral.

Cite This Article: M. Misra, B. Majhi, B.P. Padhy, P. Samanta, and U.K. Misra, “Approximation of Conjugate Series of the Fourier Series of a Function of Class W(Lp,ξ(t)) by Product Means.” American Journal of Applied Mathematics and Statistics, vol. 2, no. 5 (2014): 352-356. doi: 10.12691/ajams-2-5-9.

1. Introduction

Let na∑ be a given infinite series with the sequence

of partial sums { }ns . Let{ }np be a sequence of positive real numbers such that

0

, , 0, 0n

n i iP p n P p iυυ

− −=

= → ∞ →∞ = = ≥∑ (1.1)

The sequence –to-sequence transformation

0

1 n

nn

t p sP υ υ

υ== ∑ (1.2)

defines the sequence { }nt of the ( ), nN p -mean of the

sequence { }ns generated by the sequence of coefficient

{ }np . If

,nt s as n→ →∞ (1.3)

then the series na∑ is said to be ( ), nN p summable to s.

The conditions for regularity of ( ), nN p - summability

are easily seen to be

0

( ) , ,

( ) , .

nn

i ni

i P as n

ii p C P as n=

→ ∞ →∞ ≤ →∞

∑ (1.4)

The sequence to-sequence transformation, [1]

( ) 0

1

1

nn

n nn

T q sq

υυ

υ υ−

=

=

+∑ (1.5)

defines the sequence { }nT of the ( ),E q mean of the

sequence { }ns . If

,nT s as n→ →∞ (1.6)

then the series na∑ is said to be ( ),E q summable to s.

Clearly ( ),E q method is regular. Further, the ( ),E q

transform of the ( ), nN p transform of { }ns is defined by

( )

( )

0

0 0

1

1

1 1

1

nn k

n knk

n kn k

nkk

nq t

kq

nq p s

k Pqυ υ

υ

τ −

=

−

= =

=

+

= +

∑

∑ ∑ (1.7)

If

,n s as nτ → →∞ (1.8)

then na∑ is said to be ( )( ), , nE q N p -summable to s.

Let ( )f t be a periodic function with period 2π and integrable in the sense of Lebesgue over (-π,π). Then the Fourier series associated with f at any point x is defined by

353 American Journal of Applied Mathematics and Statistics

( )0

1 0( ) ~ cos sin ( )

2 n n nn n

af x a nx b nx A x

∞ ∞

= =+ + ≡∑ ∑ (1.9)

and the conjugate series of the Fourier series (1.9) is

( )1 1

cos sin ( )n n nn n

b nx a nx B x∞ ∞

= =− ≡∑ ∑ (1.10)

Let ( );ns f x be the n-th partial sum of (1.10). The L∞ -norm of a function :f R R→ is defined by

{ }sup ( ) :f f x x R∞ = ∈ (1.11)

and the Lυ -norm is defined by

( )

12

0

, 1f f xπ υυ

υ υ = ≥ ∫ (1.12)

The degree of approximation of a function :f R R→ by a trigonometric polynomial ( )nP x of degree n under norm . ∞ is defined by [6]

{ }sup ( ) ( ) :n nP f p x f x x R∞− = − ∈ (1.13)

and the degree of approximation ( )nE f of a function f Lυ∈ is given by

( ) minn nPn

E f P f υ= − (1.14)

This method of approximation is called Trigonometric Fourier approximation.

A function ( )f x Lipα∈ if

( )( ) ( ) ,0 1, 0f x t f x O t tα α+ − = < ≤ > (1.15)

and ( ) ( ),f x Lip rα∈ , for 0 2x π≤ ≤ , if

( )1

2

0

( ) ( ) ,

0 1, 1, 0

rrf x t f x dx O t

r t

πα

α

+ − = < ≤ ≥ >

∫ (1.16)

For a given positive increasing function ( )tξ , the

function ( ) ( )( ),f x Lip t rξ∈ , if

( ) ( ) ( )( )1

2

0

,

1, 0.

rrf x t f x dx O t

r t

πξ

+ − = ≥ >

∫ (1.17)

For a given positive increasing function ( )tξ and an

integer 1p > the function ( ) ( )( ),pf x W L tξ∈ , if

( ) ( ) ( ) ( )( )

12

0

sin ,

0.

pp pf x t f x x dx O tπ

β ξ

β

+ − =

≥

∫ (1.18)

We use the following notation throughout this paper:

{ }1( ) ( ) ( ) ,2

t f x t f x tψ = + − − (1.19)

and

( ) 0

0

1( )1

1cos cos1 2 2

sin2

n

n nk

kn k

k

nK t

kq

t tq p

tP υυ

π

υ

=

−

=

=

+

− +

∑

∑ (1.20)

Further, the method ( )( ), , nE q N p is assumed to be

regular throughout the paper.

2. Known Theorems Dealing with the degree of approximation by the

product, Misra et al [2] proved the following theorem using ( )( ), , nE q N p mean of conjugate series of Fourier

series:

2.1. Theorem If f is a 2π − periodic function of class Lipα , then

degree of approximation by the product ( )( ), , nE q N p

summability means of the conjugate series (1.10) of the Fourier series (1.9) is given by

( )1 ,0 11

n f On ατ α∞

− = < < +

where nτ is as

defined in (1.7). Recently Misra et al [3] established a theorem on

degree of approximation by the product mean ( )( ), , nE q N p of the conjugate series of Fourier series of

a function of class ( ),Lip rα . They prove:

2.2. Theorem If f is a 2π − periodic function of class ( ),Lip rα ,

then degree of approximation by the product ( )( ), , nE q N p means of the conjugate series (1.10) of the

Fourier series (1.9) is given by

( )1

1 ,0 1, 11

n

r

f O rn α

τ α∞+

− = < < ≥ +

, where nτ is

as defined in (1.7). Extending to the function of the class ( )( ),Lip t rξ ,

very recently Misra et al [4] have proved a theorem on degree of approximation by the product mean ( )( ), , nE q N p of the conjugate series of the Fourier

series of a function of class ( )( ),Lip t rξ . They prove:

American Journal of Applied Mathematics and Statistics 354

2.3. Theorem Let ( )tξ be a positive increasing function and f a

2π − Periodic function of the class

( )( ), , 1, 0Lip t r r tξ ≥ > . Then degree of approximation

by the product ( )( ), , nE q N p summability means on the

conjugate series (1.10) of the Fourier series (1.9) is given

by ( )1 11 , 1.

1rn f O n r

nτ ξ∞

− = + ≥ + , where nτ is

as defined in (1.7). Further extending to the class of functions

( )( ), , 1pW L t pξ > , in the present paper, we establish the

following theorem:

3. Main result

3.1. Theorem Let ( )tξ be a positive increasing function and f a

2π − Periodic function of the class

( )( ), , 1, 0pW L t p tξ > > . Then degree of approximation

by the product ( )( ), , nE q N p summability means on the

conjugate series (1.10) of the Fourier series (1.9) is given by

( )1 11 , 1

1rn rf O n r

nβτ ξ+ − = + ≥ +

(3.1.1),

provided

( )( )

11

1

0

sin 11

rrn t t tdt O

t n

βψξ

+ = +

∫ (3.1.2)

and

( )

( ) ( )( )1

11

1

rr

n

t tdt O n

t

δπδψ

ξ

−

+

= +

∫ (3.1.3)

hold uniformly in x with 1 1 1r s+ = , where δ is an

arbitrary number such that ( )1 1 0s δ− − > and nτ is as defined in (1.7).

4. Required Lemmas We require the following Lemmas to prove the theorem. LEMMA 4.1:

1( ) ( ) ,01nK t O n t

n= ≤ ≤

+.

Proof:

For 101

tn

≤ ≤+

, we have sin nt ≤ n sin t then

( )

( )

( )

0

0

0

0

0

1 1( ) cos cos11 2 2

sin2

cos cos .cos2 2

1sin .sin11 2

sin2

1cos

11

nn k

k

n n k

k

nn k

k

n k

k

nn k

k

n

k

nq

k

tK t tq

ptP

nq

k

t tt

ttq

ptP

nq

k

qp

P

υυ

υυ

υ

υπ

υ

υπ

π

−

=

=

−

=

=

−

=

= − ++

−

≤ ++

≤+

∑

∑

∑

∑

∑

( )

( )( ) ( )( )

( )

2

0

0

0

0 0

0

2sin2 2 sin

sin2

1

112sin sin sin

2 2

1 1

1

1

1

k

nn k

kn k

k

n kn k

nkk

nn k

nk

t t

tt

nq

k

t tqp O t

P

nq p O O

k Pq

n Oq

kq

υ

υυ

υυ

υυ

πυ υ υ

υ υπ

π

=

−

=

=

−

= =

−

=

+

≤+

+

≤ ++

≤+

∑

∑

∑

∑ ∑

∑0

( )

( )

k

k

kp

P

O n

υυ=

=

∑

This proves the lemma. LEMMA 4.2:

1 1( ) ,1nK t O for t

t nπ = ≤ ≤ +

.

Proof:

For 11

tn

π≤ ≤+

, by Jordan’s lemma, we

have sin2t t

π ≥

.

Then

( )

0

0

1 1( ) cos cos11 2 2

sin2

nn k

k

n n k

k

nq

k

tK t tq p

tP υυ

υπ

−

=

=

= − + +

∑

∑

355 American Journal of Applied Mathematics and Statistics

( )

( )

( )

0

0

0

2

0

cos cos .cos2 2

1sin .sin11 2

sin2

1cos 2sin

11 2 22

sin .sin2 2

2 1

nn k

k

n k

k

nn k

k

n k

k

n

nq

k

t tt

ttq ptP

nq

k

t tq p

P t t t

nkq t

υυ

υυ

υ

υπ

υππ

υ

π

π

−

=

=

−

=

=

−

= ++

≤ + +

≤

+

∑

∑

∑

∑

( )

( )

0 0

0 0

0

1

1 1

2 1

1

2 1

1

n kn k

kk

n kn k

nkk

nn k

nk

q pP

nq p

k Pq t

nq

kq t

Ot

υυ

υυ

−

= =

−

= =

−

=

= +

=

+

=

∑ ∑

∑ ∑

∑

This proves the lemma.

5. Proof of Main Theorem Using Riemann–Lebesgue theorem, for the n-th partial

sum ( );ns f x of the conjugate Fourier series (1.10) of ( )f x and following Titchmarch [5], we have

( )0

1cos sin

2 2 2; ( ) ( ) ,2sin

2

n

tn t

s f x f x t dtt

π

ψπ

− +− =

∫

Using (1.2), the ( ), nN p transform of ( );ns f x is given by

00

1cos sin2 2 2( ) ( )

2sin2

n

n kn k

t n tt f x t p dt

tP

πψ

π =

− + − =

∑∫ ,

Denoting the ( )( ), , nE q N p transform of ( );ns f x by

nτ , we have

( )

0

0

0

11

10 01

1 2

( )

2 1cos sin11 2 2

2sin2

( ) ( ) ( ) ( )

,

nn

n k

k

n k

k

n

n n

n

f

nt q

k

t dttq p

tP

t K t dt t K t dt

I I say

π

υυ

π π

τ

ψ

υπ

ψ ψ

−

=

=

+

+

− =

− + +

= = +

= +

∑

∫∑

∫ ∫ ∫

(5.1)

Now

( )

( )( )

( ) ( )

1

011

0

0

11

0

1 11 1

1 1

0 0

( )

2 1cos cos11 2 2

2sin2

( ) ( )

sin

sin

nn k

kn

n k

k

n

n

r sr sn nn

I

nt q

k

t dttq p

tP

t K t dt

t t t t K tdt dt

t t t

υυ

β

β

ψ

υπ

ψ

ψ ξξ

−

=+

=

+

+ +

=

− + +

≤

≤

∑

∫∑

∫

∫ ∫

where 1 1 1r s+ = , using Hölder’s inequality

( )

( )

( )

( )

( )

11

1

10

11

1

10

1 1

1

(1) ,

1( )

4.1 3.1

1

11

2

1

.

1

11

ssn

sn

s

s

r

using Le

tO dt

t

dtOn t

O O n

mma

n

d

n

O

an

n

β

β

β

β

ξ

ξ

ξ

ξ

+

+

+

+

− + +

+

=

= +

= + + = + +

∫

∫ (5.2)

Next

American Journal of Applied Mathematics and Statistics 356

( )( )

( ) ( )

1 1

21 1

1 1

sin

sin

r sr sn

n n

t t t t K tI dt dt

t t t

π δ β π

δ βψ ξξ

−

−

+ +

≤

∫ ∫

where 1 1 1r s+ = , using Hölder’s inequality

( ) ( )

1

11

1

( 1 )

ss

n

tO n dt

t

πδ

β δξ+ −

+

= +

∫ , using Lemma 4.2 and

(3.1.3) ( )

1

1

1 21

1

( 1 )

s s

n y dyO ny y

δδ β

π

ξ+

− −

= +

∫

since ( )tξ is a positive increasing function, so is

( ) ( )1/ / 1/y yξ . Using second mean value theorem we get

( ) ( )

( ) ( )

( )

11

11 2

11 1

1

1( 1 ) ,1

1 1

11 11

111

n s

s

s

r

dyO nn y

for some n

O n O nn

O nn

δδ β

ε

δ β δ

β

ξ

επ

ξ

ξ

++

− − +

+ + − −

+

= + +

≤ ≤ +

+ + + = + +

∫

(5.3)

Then from (5.2) and (5.3), we have

( ) ( )

( ) ( )

( )

( )

1

12 1

0

121

0

1

11 , 11

11 , 1.1

111

111

rn

r rrn r

rr

r

f x O n for rn

f x O n dx rn

O n dxn

O nn

β

πβ

πβ

β

τ ξ

τ ξ

ξ

ξ

+

+

+

+

− = + ≥ +

− = + ≥ +

= + + = + +

∫

∫

This completes the proof of the theorem.

6. Corollaries Following corollaries can be derived from the main

theorem. Corollary 6.1: The degree of approximation of a

function f belonging to the class

( ), ,0 1, 1Lip r rα α< ≤ ≥ is given by

( )1

1 .rn rf O n ατ − + − = +

Proof: The corollary follows by putting 0β = and

( )t tαξ = in the main theorem. Corollary 6.2: The degree of approximation of a

function f belonging to the class ( ) ,0 1Lip α α< ≤ is given by

( )( )1 .n f O n ατ −∞− = +

Proof: The corollary follows by letting r →∞ in corollary 6.1.

References [1] G.H. Hardy, Divergent Series (First Edition), Oxford University

Press, (1970). [2] U.K. Misra, M. Misra, B.P. Padhy and S.K. Buxi, “On degree of

approximation by product means of conjugate series of Fourier series”, International Jour. of Math. Scie. And Engg. Appls. ISSN 0973-9424, Vol 6 No. 1 (Jan. 2012), pp 363-370

[3] Misra U.K.,Paikray, S.K., Jati, R.K, and Sahoo, N.C.: “On degree of Approximation by product means of conjugate series of Fourier series”, Bulletin of Society for Mathematical Services and Standards ISSN 2277-8020, Vol. 1 No. 4 (2012), pp 12-20.

[4] U.K. Misra, M. Misra, B.P. Padhy and D.Bisoyi, “On Degree of Approximation of conjugate series of a Fourier series by product summability" Malaya Journal of Mathematik (ISSN: 2319-3786, Malayesia), Vol. 1 Issue 1 (2013), pp 37-42.

[5] E.C. Titchmarch, The Theory of Functions, Oxford University Press, (1939).

[6] A. Zygmund, Trigonometric Series (Second Edition) (Vol. I), Cambridge University Press, Cambridge, (1959).