CONVERGENCE ANALYSIS OF SOME APPROXIMATION OPERATORS

Abstract

The thesis is divided into seven chapters, the contents of which are organized as follows: Chapter1 of the thesis covers the literature and historical foundation of certain important approximation operators. We provide a brief overview of the chapters that constitute this thesis and discuss some of the preliminary tools we will use to delve into the subject’s depth. Chapter 2 introduces a new sequence of operators involving Apostol-Genocchi polynomials and Baskakov operators and their integral variants. We estimate some direct convergence re- sults using the second-order modulus of continuity, Voronovskaja type approximation theorem. Moreover, we find weighted approximation results of these operators. Next Chapter 3 is mainly focused on the difference operators of two positive linear operators (generalized pˇaltˇanea type operators Lλ n,c ( f ; x) and M. Heilmann type operators Mn,c( f ; x) ) with same basis functions. First, we estimate quantitative difference of these operators in terms of modulus of continuity and Peetre’s K−functional In Chapter 4, we present a recurrence relation for the semi-exponential Post-Widder operators and provide estimates for their moments. We then examine convergence results within Lipschitz- type spaces, analyzing the convergence rate using the Ditzian-Totik modulus of smoothness and the weighted modulus of continuity. Finally, we estimate the convergence rate for functions whose derivatives are of bounded variation. Chapter 5 introduces a novel Bézier variant within the family of Phillips-type generalized positive linear operators. The moments of these operators are derived to enhance understand- ing of their fundamental properties. The chapter further explores convergence properties in Lipschitz-type spaces, with particular focus on the Ditzian-Totik modulus of smoothness. Fi- nally, it provides a rigorous analysis of the convergence rate for functions whose derivatives are of bounded variation, contributing valuable insights to the field of approximation theory. The aim of Chapter 6 is to introduce the sequence of Baskakov-Durrmeyer type operators linked with the generating functions of Boas-Buck type polynomials. After calculating the mo- ments, including the limiting case of central moments of the constructed sequence of operators, in the subsequent sections, we estimate the convergence rate using the modulus of continuity and Ditzian-Totik modulus of smoothness and some convergence results in Lipchitz-type space and the end we estimates the convergence for the functions of bounded variations. The thesis is summarised in Chapter 7, before providing some insight into the author’s thoughts about the future research.