Absolute indexed summability methods and approximation theorems

Abstract

Over the years, the absolute indexed summability methods and the theory approximation newlinehave gained considerable importance, and several works have been appeared newlinein the literature using various types of summability methods (ordinary and statistical newlineversions), namely Ces`aro summability, Riesz summability, N¨orlund summability, Euler newlinesummability, etc. In this thesis, some advancement has been made by the authors with newlinethe introduction of several new results by using absolute indexed Ces`aro and N¨orlund newlinesummability, deferred Riesz, deferred Euler, deferred Weighted A-statistical and deferred newlineN¨orlund-Euler product summability means. Various results established via the newlineabove-mentioned means are briefly discussed in different chapters. newlineThe present thesis comprises eight chapters appended with a bibliography. Chapter newline1 deals with a brief introduction and literature survey on different summability methods newlineassociated with our present work. In Chapter 2, we have studied the Ces`aro summability newlineof improper integrals to establish a new theorem via absolute indexed Ces`aro newlinesummability factors of improper integral. In Chapter 3, we have studied the |N, pn, and#948;|rsummability newlineof improper integrals to establish a new result under minimal sufficient newlineconditions. In Chapter 4, we have studied the notion of statistical deferred weighted newline(Riesz) summability means for trigonometrical periodic functions to establish a new newlineKorovkin-type approximation theorem. Moreover, we pursued the rate of statistical deferred newlineweighted summability and established another result for the same set of functions newlinewith the help of the modulus of continuity. In Chapter-5, we have considered deferred newlineEuler statistical convergence as well as statistical deferred Euler summability methods newlineto obtain a Korovokin-type Approximation Theorem. In Chapter 6, we have introduced newlinethe concept of statistical deferred weighted A-summability and deferred weighted Astatistical newlineconvergence concerning the difference sequence of order r involving (p, q)- newlineintegers.