7/29/2019 Aw 31326337 http://slidepdf.com/reader/full/aw-31326337 1/12 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.326-337326 | P age Degree Of Approximation Of Functions By Modified Partial Sum Of Their Conjugate Fourier Series By Generalized Matrix Mean Prakash Chandra Rautaray 1 , Ellipse 2 1 Department of Mathematics, KIIT University, Bhubaneswar-751024, Odisha, India2 Department of Mathematics, Maharishi College of Natural Law, Bhubaneswar -751007, Odisha, India Abstract The 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 by generalized matrix mean in generalized Holder metric. Keyword. Banach Space, generalized Holder metric and regular generalized matrix. 1. Definition and Notation The 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 1 2 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 , sup c 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 p h 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 sup p 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 sup c 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 complete in the metric defined by its norm is called Banach Space. (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 p h 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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7/29/2019 Aw 31326337
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Prakash Chandra Rautaray, Ellipse / International Journal of Engineering Research and
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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Prakash Chandra Rautaray, Ellipse / International Journal of Engineering Research and
Vol. 3, Issue 1, January -February 2013, pp.326-337
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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Prakash Chandra Rautaray, Ellipse / International Journal of Engineering Research and