Degree of Approximation and Absolute Index Summability Of Series Of Real and Fuzzy Numbers

Abstract

Over the years, the absolute indexed summability methods and the theory approx- imation have gained considerable importance, and several works have been appeared in the literature using various types of summability methods (ordinary and statistical versions) for sequences of real and fuzzy numbers, namely Ces`aro summability, Riesz summability, N¨orlund summability, Euler summability, etc. In this thesis, some ad- vancement has been made by the author with the introduction of several new results based on Euler-Matrix, Euler-Hausdorff, Euler-Riesz (N¨orlund type) and Ces`aro-Euler composite product summability means. Moreover, statistical deferred weighted and de- ferred N¨orlund summability means are used to establish some approximation theorems. Various results established via the above-mentioned means are briefly discussed in dif- ferent chapters. newlineThe present thesis comprises eight chapters appended with a bibliography. Chapter 1, deals with a brief introduction and literature survey based on various summability methods associated with our present work. In Chapter 2, we have established two the- orems to estimate the degree of approximation of Fourier series and conjugate Fourier series via (E,q)A- product means in the generalized Lipschitz Lip(and#958;(t),r) class. In Chapter 3, we have considered the Euler-Hausdroff product summability mean to esti- mate the degree of approximation of Fourier series in the weighted Zygmund W(Z(and#969;) r ) class. In Chapter 4, we have introduced (E,1)(N,pn)- product summability mean in the weighted Zygmund W(Z(and#969;) r ) class to estimate the degree of approximation of conjugate Fourier series. In Chapter 5, we have studied the deferred weighted mean and established an inclusion relation connecting deferred weighted statistical convergence and statistical deferred weighted summability. Moreover, we established a Korovokin-type approxima- tion theorem for algebraic test functions.