Some aspects on recent development of summability theory and its applications to fourier series

Abstract

The thesis entitled SOME ASPECTS ON RECENT newlineDEVELOPMENT OF SUMMABILITY THEORY AND ITS newlineAPPLICATIONS TO FOURIER SERIES under the joint supervision newlineof Dr. Mahendra Misra and Dr.Padmanava Samanta, is embodied newlinewith seven chapters. The first chapter is the introduction and newlineliterature survey on the related work where the background newlineliterature has been described briefly in the light of which the newlineinvestigation of the author have been presented in the subsequent newlinechapters. Chapter 2 to Chapter 7 deals with the contribution of the author in the field of summability theory and its application to newlineFourier series. The bibliography section and the information about newlinethe research publications of the papers are presented at the end. newlineThe chapter wise summary of the proposed work is given newlinebelow: newlineBernestein, Alexits, Sahney and Goel, Chandra and several newlineothers have determined the degree of approximation of the Fourier newlineseries of the function f and#61646;Lipand#61537; by and#61480; and#61481; and#61480; and#61481; and#61480; and#61481; n C,1 , C,and#61540; , N, p and and#61480; and#61481; n N, p newlinemeans. Subsequently, working on the same direction Sahney and Rao, and Khan have established results on the degree of newlineapproximation of the function belonging to the class Lip and#61537; and newlineLip and#61480;and#61537;, rand#61481; by and#61480; and#61481; n N, p and and#61480; and#61481; n n N, p , q means respectively. However, newlinedealing with product summability Nigam and Sharma proved a theorem on the degree of approximation by the product and#61480; and#61481;and#61480; and#61481; 1, , CqE - newlinemean of Fourier series. Subsequently Misra et al have established a newlineon degree of approximation by the product mean and#61480; and#61481;and#61480; and#61481; n E,q N, p of the newlineFourier series.