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Prakash Chandra Rautaray, Ellipse / International Journal of Engineering Research and

Applications (IJERA) ISSN: 2248-9622 www.ijera.com

Vol. 3, Issue 1, January -February 2013, pp.326-337

326 | P a g e

Degree Of Approximation Of Functions By Modified Partial Sum

Of Their Conjugate Fourier Series By Generalized Matrix Mean

Prakash Chandra Rautaray1, Ellipse

2

1

Department of Mathematics, KIIT University, Bhubaneswar-751024, Odisha, India 2Department of Mathematics, Maharishi College of Natural Law, Bhubaneswar -751007, Odisha, India

AbstractThe paper studies the degree of

approximation of conjugate of a 2 -periodic

Lebesgue integrable function f by using modified

partial sum of its conjugate Fourier series bygeneralized matrix mean in generalized Holder metric.

Keyword. Banach Space, generalized Holder metric and regular generalized matrix.

1. Definition and NotationThe following definitions will be used

throughout the paper (see Zygmund [8] p.16,42, [5]

p.2,49 and [3] ).

(i) The space , p L includes the space

of all 2 -periodic Lebesgue integrable continuous

functions defined in , with p-norm given by

12

0

2

0

sup ;

; 1

;0 1 .

t

p p

p

p

f t p

f f t dt p

f t dt p

(ii) 0

, supc

h

w w f f x h f x

when p , is called the modulus of continuity

0

, sup p p p

h

w w f f x h f x

is called the integral modulus of continuity.

2 2

0

,

sup 2

p p

ph

w w f

f x h f x h f x

is called the integral modulus of smoothness.(iii) The Lipschitz condition is given by

or

0

supp

h

f x h f x K

(+ve constant )

when 0 . p

(iv) The Holder metric space H is defined by

2 : ; 0,0 1 H f C f x f y K x y K

with Holder metric induced by the norm

,

sup , sup supc

x y t x y

f x f y f f f x y f t

x y

where

, f x f y

f x y x y

and 0 for

0 .

(v) A normed linear space which is completein the metric defined by its norm is called BanachSpace.

(vi) The generalized Holder metric space

, H p is defined by

, : p p H p f L f x h f x K h

where K > 0(constant), 0< 1 and 0 . p

Also the metric given by

,

sup , sup p

p p ph h

f x h f x f f f x h x f

h

and ,o p p

f f for 0 , is called generalized

Holder metric.

(vii) , H p is a complete normed linear space

and hence a Banach space for 0 1. p

Also , . H H

0

sup c

h

f x h f x K

(+ve constant) when p

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327 | P a g e

(viii)A generalized matrix nk M m i is said to be regular if

, 0

sup nk n i k

M m i

and 0

1nk

k

m i

as n uniformly in i.

(ix) Let , ; 1, p f L p be a 2 -periodic Lebesgue integrable function.

Then the conjugate Fourier series of f at ,t x is given by

1

sin cos ;k k

k

f x a kx b kx

(1)

where 1

cosk a f t kt dt

and 1

sin .k b f t kt dt

The following notations shall be used throughout the paper.

*

1 cos

2tan2

n

nt D t

t

is called Dirichlet’s modified conjugate kernel.

1

2 x t f x t f x t

, , 1

0

n n k n k

k

i m i m i

0

1 cos;

2tan2

n nk

k

kt K i t m i

t

(2)

*

0

cos;2tan

2

n nk

k

kt H i t m it

(3)

0

2

2tan2

xt

f x dt t

is called conjugate function of f x .

2

, ,2tan

2n

x

n

t f x f x dt

t

; (4)

where 0n

is very small positive number.

* *

0

2n x n

S x t D t dt

(5)

is called modified nth partial sum of conjugate Fourier series of f x given by (1).

* *

0

n nk k

k

M S x m i S x

(6)

uniformly in i, provided the series exists for each n, which is called the matrix

transformation of *

nS x .

* *; ,n n

n

l i x M S x f x

(7)

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328 | P a g e

2. IntroductionChandra [1] and Sahney have determined

the degree of approximation of a function

belonging to Lip by ,1 , ,C C and

, n N p means. In 1981, Quereshi discussed the

degree of approximation of conjugate of a function

belonging to Lip and Lip , p by , n N p

means of con jugate series of a Fourier series. In2000, Shyam Lal [4] determined the degree of

approximation of conjugate of function belonging

to weighted class , pW L t by matrix means

of conjugate series of Fourier series. Also in 2001,G.Das, R.N.Das and B.K.Ray[3] studied the degreeof approximation in same direction using infinitematrix mean in generalized Holder metric.

The objective of the present paper is tostudy more comprehensively the result of G.Das,R.N.Das & B.K.Ray[3] by generalized

matrix mean.

3. Result In this paper we have studied the degree of

approximation of conjugate function of f x by

modified partial sum of its conjugate Fourier series

by generalized matrix mean in generalized Holder metric i.e.

* *

,0

,

; ,n nk k p

k n p

l i x m i S x f x

uniformly in i.

The following lemma will be required for establishing the theorem.

Lemma.

Let 0 p .

Then (a) , x p pt w f

and (b) x y x pt t K

p f x t y f x t ;where K > 0 (constant).

Proof .

(a) For 1 p and by Minkowski’s inequality, we have

1 12 2 2

0 0 0

p p p p p f x t f x t dx f x t f x dx f x f x t dx

and for 0 1 p , we have by modified Minkowski’s inequality

2 2 2

0 0 0

p p p f x t f x t dx f x t f x dx f x f x t dx

0

sup p

t

f x t f x t

0 0

sup sup p p

t t

f x t f x f x f x t

0

2 sup p

t

f x t f x

0 0

sup sup2 p

t t p

f x t f x t f x t f x

, x p pt w f

(b) Now

x y xt t = 1

2 f x y t f x y t f x t f x t

= 1 1

2 2 f x t y f x t f x t f x t y

By Minkowski’s inequality for 1 p and 0 1 p separately, we get

1 1

2 2 x y x p p pt t f x t y f x t f x t f x t y

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329 | P a g e

p

K f x t y f x t ;where K > 0 is a constant

which establishes the lemma.

Theorem.

If for 1,0 1 p and for positive increasing sequences n and n such that n n and

nk M m i (space of regular matrices) such that 2

1n

nk n

k

k m i O

and , f H p , then

*

1,

1

1log

; 11 ;0 1

log log ; 1

nn n

n n

n p

n n

n

n n n

l i x O

i

uniformly in i.

Proof.

The equation (7) can be written as

* *; ,n n

n

l i x M S x f x

= *

0

,nk k

k n

m i S x f x

=

0 00

1 cos2 2

2 tan 2 tan2 2n

x

nk x nk

k k

kt t m i t dt m i dt

t t

( 0

1nk

k

m i

as n )

=

0 00

1 cos2 2 1

2 tan 2 tan

2 2n

x nk x nk

k k

kt t m i dt t m i dt

t t

(provided the change of order of summation is permitted)

=

0 00

1 cos cos2 2

2 tan 2 tan2 2

n

n

x nk x nk

k k

kt kt t m i dt t m i dt

t t

*

0

2 2; ;

n

n

x n x nt K i t dt t H i t dt

(8)

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330 | P a g e

Now

0

1 cos;

2tan2

n nk

k

kt K i t m i

t

0

2 1 cos 2

2tan2

nk

k

m i kt t

0

22 tan 0

2nk

k

t m i t for t

t

2 M

t

1O t

1;n K i t O t (9)

Also for 1,0 1 p and , , f H p

p f x t f x O t

(10)

By lemma(a), 0

x p pt

t Sup f x t f x O t

( by (10))

x pt O t

(11)

Again x y x pt t

x y x p pt t

(by Minkowski’s inequality )

O t O t

( by (11))

x y x p

t t O t

(12)

Consider * *; ;n n

pl i x y l i x

= *

0

2 2; ;

n

n

x y x n x y x n

p

t t K i t dt t t H i t dt

*

0

2 2

; ;

n

n

x y x n x y x n p pt t K i t dt t t H i t dt

= 1 2 I I (say) (13)

where 1

0

2;

n

x y x n p I t t K i t dt

= 1

0

2 n

O t O t dt

(by (12) and (9))

= 1

0

1n

O t dt

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331 | P a g e

= 0

1nt

O

= 1

1n

O

(14)

Also by Abel’s transformation for n

t

,

1

, , 1 ,

0 0

0

cos lim ;

1sin cos

2 2

sin2

n

nk k n k n k n n nn

k k

n

n

k

m i kt S m i m i S m i

n nt t

where S cos kt t

(see[2],[8])

, , 1

0

1sin cos2 2

sin2

n k n k

k

k kt t

m i m it

as ,lim 0n nn

m i

, , 1

0 0

1cos

sin2

nk n k n k

k k

m i kt m i m it

1sin cos

12 2

sin sin2 2

k kt t

t t

= 1

, , 1

0

n k n k

k

O t m i m i

= 1

nO t i

1

0

cosnk n

k

m i kt O t i

for

n

t

(15)

and *

2

2;

n

x y x n p I t t H i t dt

=

0

cos2

2tan2n

nk

k

kt O t m i dt

t

( by (12))

= 1

0

2cos

n

nk

k

O t O t m i kt dt

2 tan 0

2

t t for t

= 1 12

n

nO t O t O t i dt

(by (15)

= 21

n

nO i t dt

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332 | P a g e

=

1

;0 11 1

log ; 1n

n

t

O i

t

=

1;0 1

1log ; 1

nn

n

O i

(16)

Further order estimates for I1 and I2 can be obtained as follows

1

0

2 2; ;

n n

n

x y x n x y x n p p I t t K i t dt t t K i t dt

= 11 12 I I (say) (17)

Also

0

1 cos;

2tan 2

n nk

k

kt K i t m i

t

2 2

0 1

sin sin2 2

tan tan2 2

n

n

nk nk

k k

kt kt

m i m it t

2 2 2 2

0 14 tan 4 tan2 2

n

n

nk nk

k k

k t k t m i m i

t t

sin2 2

kt kt

2 2

0 12 2

n

n

nk n nk

k k

t t m i m i k

2tan2

t t

= 2 2

0 1

n

n

nk n nk

k k

O t m i k m i

= n nO t O O ( 0

nk

k

m i

and by necessary condition of theorem)

;n n K i t O t O (18)

Again by lemma (b), x y x pt t O y

(19)

Now 11

0

2;

n


= 0

2 n

nO y O O t dt

( by (18),(19) )

= 0

n

nO y O tdt

=

2

02

n

n

t O y O

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333 | P a g e

= 2

1n

n

O y O O

= 1

n

O y

(20)

and 12

2;

n

n


=

0

1 cos2

2tan2

n

n

nk

k

kt O y m i dt

t

( by (19))

=

1 2

0

2sin 2 tan

2 2

n

n

nk

k

kt t O y O t m i dt t

= 1 2

0

2 sin 12

n

n

nk

k

kt O y O t m i dt

= 1n

n

O y M O t dt

= 1

n

n

O y dt t

= log n

n

O y t

log n

n

O y

(21)

Hence 1 11 12

1log n

n n

I I I O y

(22)

Combining (14) and (22), we get for 0 1

1

1 1 1 I I I

= 1

1 1log 1n

n n n

O y O

1log n

n

n n

O y

(23)

Also 2

2,

n


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334 | P a g e

= 1

0

2

n

nk

k

O y O t m i cos kt dt

( by (19) and 2tan

2

t t )

= 1 1

n

nO y O t O t i dt

( by (15) )

= 2

n

nO y i t dt

= 2

n

nO y i t dt

= 1

1n

n

t O y i

= n n i O y (24)

Combining (16), (24), we have for 0 1

1

2 2 2 I I I

=

111 ;0 1

1 log ; 1

n n

n n

n n

O ii O y

O i

=

1

1

;0 1

log ; 1

n

n

n n

O y i

(25)

Hence * *1 2

; ;n n

pl i x y l i x I I

=

1

1

;0 11log

log ; 1

nnn n

n n n n

O y O y i

* *

1, 0

1

1log

; ;sup 1

;0 1

log ; 1

nn

n n p n n

n yn

n

n n

l i x y l i xO

yi

(26)

Further * ;n

pl i x

*

0

2 2; ;

n

n

x n x n p pt K i t dt t H i t dt

= 1

00

cos2 2

2tan2

n

n

nk

k

kt O t O t dt O t m i dt

t

( using (11) in both integrals & (9) in 1st

integral )

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335 | P a g e

= 1 1

00

1 1 cos 2 tan2

n

n

nk

k

t O t dt O t O t m i kt dt t

= 1 1

0

1 1n

n

nt O O t O t i dt

( by (15) )

= 211 1

n

n

n

O O i t dt

=

1

;0 11 1 1

log ; 1n

n n

t

O O i

t

=

1 ;0 11 1

log ; 1nn n

n

O O i

1

* ;0 1; 1

log ; 1

n

n n n p

n

l i x O i

(27)

Adding (26) and (27), we have

* *

* *

, , 0

; ;; ; sup

n n p

n n p p n y

l i x y l i xl i x l i x

y

1

1

1 ;0 11

1 loglog log ; 1

n nn

n n n

n nn n n

O i

(28)

Hence the result follows.

This completes the proof of theorem.

4. CorollariesUsing the above theorem the following two corollaries can be established.

Corollary1

If for 1,0 1, , p f H p and nk M m i is a lower triangular matrix of non-negative real

numbers with monotonic increasing in k such that 0

1n

nk

k

m i

as n uniformly in i, then

1

*

, 1,

1 ;0 1; 1 1

log log ; 1n n n

p

n nl i x O n n m i

n n n

Proof.

Let 1,0 1, , . p f H p

Let nk M m i be a lower triangular matrix of non-negative real numbers with monotonic increasing in k

such that 0

1n

nk

k

m i

as n uniformly in i i.e.

, , 10, 0,1, 2..., 1nk n k n k m i m i m i k n and 0

1

n

nk

k m i


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336 | P a g e

Choose .n n n

Then 2 2

1 1n

nk nk

k k n

k m i k m i

= 2 2

1 0 2 0 ...n n

= O n (29)

Also , , 1 , 1 ,

0 0

n

n n k n k n k n k

k k

i m i m i m i m i

= 1

, 1 , , 1 ,

0

n

n k n k n n n n

k

m i m i m i m i

= 1

, 1 , , , 1

0

0n

n k n k n n n n

k

m i m i m i m i

= ,1 ,0 ,2 ,1 , , 1 ,...n n n n n n n n n nm i m i m i m i m i m i m i

= ,0 , ,n n n n nm i m i m i

= , ,02

n n nm i m i

,2 n nm i

,n n ni O m i (30)

Clearly all conditions of theorem are hold.

Hence

1

*

, 1,

1 ;0 1; 1

log log ; 1n n n

p

n nl i x O n n n O m i

n n n

1

, 1

1 ;0 11 1

log log ; 1n n

n nO n n m i

n n n

(31)

This establishes corollary 1.

Corollary 2

If for 1,0 1, , p f H p and nk M m i is a lower triangular matrix of non-negative real

numbers with monotonic decreasing in k such that 0

1n

nk k

m i

as n uniformly in i, then

1

*

,0 1,

1 ;0 1; 1 1

log log ; 1n n

p

n nl i x O n n m i

n n n

Proof.

Let 1,0 1, , . p f H p

Let nk M m i be a lower triangular matrix of non-negative real numbers with monotonic decreasing in k

such that 0

1n

nk k

m i

as n uniformly in i i.e.

, , 10, 0,1, 2..., 1nk n k n k m i m i m i k n and 0

1n

nk k

m i


Choose .n n n

Then 2 2

1 1n

nk nk

k k n

k m i k m i

= 2 2

1 0 2 0 ...n n

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337 | P a g e

= O n (32)

Also , , 1

0

n n k n k

k

i m i m i

, , 1

0

n

n k n k

k

m i m i

, , 1

0

n

n k n k

k

m i m i

= ,0 ,1 ,1 ,2 , , 1 , 1... 0n n n n n n n n n nm i m i m i m i m i m i m i

= ,0nm i

,0n ni m i (33)

Clearly all conditions of theorem are hold.

Hence

1

*

,0 1,

1 ;0 1; 1

log log ; 1n n

p

n nl i x O n n n m i

n n n

1

,0 1

1 ;0 11 1

log log ; 1n

n nO n n m i

n n n

(34)

This establishes corollary 2.

Remark The above result improves the result of G.

Das, R. N. Das and B. K. Ray [3] taking modified

partial sum *

nS x of conjugate Fourier series of

f x in place of nS x (see [8]).

References[1] Prem Chandra, On degree of

approximation of functions belonging to

the Lipschitz class, Nanta math. 8(1975),88-89

[2] G.Das, S.Pattanayak, Fundamentals of

mathematical analysis (1987), TataMCGraw-Hill Publishing companylimited, New Delhi

[3] G. Das, R.N. Das &B.K. Ray, Degree of

approximation of functions by their conjugate Fourier series in generalized

Holder metric, Journal of the Orissamathematical society, vol.17-20, 1998-2001, pp.61-74

[4] Shyam Lal, On degree of approximationof conjugate of a function belonging to

weighted , pW L t class by matrix

sumability means of conjugate series of a

Fourier series, Tamkang Journal of mathematics (2000)

[5] G.G.Lorentz(SyrocuseUniversity),

Approximation of functions (1948),(Athera series, Edwin Hewill, Editor)

[6] Kutbuddin Qureshi, On the degree of approximation of functions belonging to

the class Lip , p by means of

conjugate series, Indian J. Pure Applied,4(1981)

[7] E.C Titchmarsh (University of Oxford), Atheory of functions (1939), (Oxford

University Press, New York)

[8] A. Zygmund,Trigonometric series (1977)(Cambridge University Press, New York)vol.I

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