IJPAM: Volume 70, No. 3 (2011)

ON APPROXIMATION OF CONJUGATE OF A FUNCTION
BELONGING TO WEIGHTED $W\left({{L_r},\xi \left( t \right)} \right)$
CLASS BY PRODUCT MEANS

H.K. Nigam$^1$, Ajay Sharma$^2$
Department of Mathematics
Faculty of Engineering and Technology
Mody Institute of Technology and Science (Deemed University)
Laxmangarh, 332311, Sikar (Rajasthan), INDIA


Abstract. A good amount of work has been done on degree of approximation of functions belonging to $Lip\alpha , Lip\left( {\alpha ,r} \right), Lip\left({\xi \left( t \right),r} \right)$ and $W\left({{L_r},\xi \left( t \right)} \right)$ classes using Ces$\grave{a}$ro, N$\ddot{o}$rlund and generalised N$\ddot{o}$rlund single summability methods by a number of researchers (see [#!ga!#], [#!pc!#], [#!hk!#], [#!ll!#], [#!kq1!#], [#!kq2!#], [#!qn!#], [#!br!#], [#!sg!#]). But till now, nothing seems to have been done so far to obtain the degree of approximation of conjugate of a function using $(N,p_{n})(C,1)$ product summability method of its conjugate Fourier series. Therefore, the purpose of present paper is to establish a quite new theorem on degree of approximation of a function $\tilde{f}$, conjugate to a $2\pi-$periodic function $f$ belonging to weighted, i.e. $W\left({{L_r},\xi \left( t \right)} \right)$ class, $r\geq1$, by $(N,p_{n})(C,1)$ product summability means of its conjugate Fourier series.

Received: December 28, 2010

AMS Subject Classification: 42B05, 42B08

Key Words and Phrases: degree of approximation, $W\left({{L_r},\xi \left( t \right)} \right)$ class of function, $(N,p_{n})$ mean, $(C,1)$ mean, $(N,p_{n})(C,1)$ product means, Fourier series, conjugate Fourier series, Lebesgue integral

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2011
Volume: 70
Issue: 3