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I nternational Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 7

Lp  convergence of Rees-Stanojevic sum Nawneet Hooda Depart ment of Mathemat ics, DCR Un iversity of Sci& Tech, Murthal , Sonepat-131039 (INDIA)

Abstract We study Lp -convergence (0<p<1) of Rees -Stanojevic modified cosine sum[3] and deduce the result of Ul’yanov [4] as corollary fro m our result.

1. Introduction Let us consider the series (1.1)

f(x) =

a0 2

+

 a k cos kx ,

k 1

with coefficients a k  0 or even satisfying the conditions ak  0 as k and

k 1

|ak | <  . Riesz [Cf .1] showed that

the function f(x) defined by the series (1.1) for ak  0 can be non-summable. However, they are summab le to any degree p provided 0 < p < 1. Theorem A.[4] If the sequence < a k > satisfies the condition a k  0 and  |ak | < + , then for any p, 0 < p < 1 , we have 

limn

p

 f ( x)  S

n

( x) dx = 0 ,

where Sn (x) is the partial sum o f the series (1.1) . Rees and Stanojevic [3] (see also Garrett and Stanojevic [ 2 ] ) introduced a modified cosine sum (1.2)

h n (x) =

1 n  2 k 0

n

ak + 

k 1

n

j k

aj cos kx .

Regarding the convergence of (1.2) in L-metric, Garrett and Stanojevic [2] proved the follo wing result : Theorem B. If {ak } is a null quasi-convex sequence . Then ||h n (x)  f(x)|| = o(1), n , where f(x) is the sum of cosine series (1.1). In this paper ,we study the Lp convergence of this modified sum (1.2) and deduce Theorem A as corollary of our theorem.

2000 AMS Mathematics Subject Classification : 42A20, 42A32.

Issn 2250-3005(online)

November| 2012

Page

45


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