ANALYSIS OF CERTAIN APPROXIMATION OPERATORS

Abstract

The thesis is divided into seven chapters, the contents which are organized as follows: Chapter1 of the thesis covers the literature and historical foundation of certain important approximation operators. We give a brief overview of the chapters that make up this thesis and talk about some of the preliminary tools we’ll use to get to the subject depth. Chapter 2 introduces a new sequence of operators involving Apostol-Genocchi polynomials and their integral variants. We estimate some direct convergence results using the second-order modulus of continuity, Voronovskaja type approximation theorem. Moreover, we find weighted approximation results of these operators. Finally, we derive the Kantorovich variant of the given operators involving Apostol-Genocchi polynomials, and their approximation properties are studied. Next Chapter 3 is mainly focused on the Bézier variant of the Bernstein-Durrmeyer type operators. First, we estimate the moments for these operators. Then, we determine the rate of approximation of the operators R˘ (ρ,α) n,r,s (f ; x) in terms of the Ditzian-Totik modulus of continuity and over the Lipschitz-type spaces. It is addressed how smooth functions with derivatives of bounded variation converge. Finally, graphic depiction of the theoretical findings and the efficiency of these operators are shown. Chapter 4 deals with certain approximation properties of Cheney-Sharma Chlodovsky Dur rmeyer operators. Using the moments of these operators Bohman-Korovkin’s theorem is val idated. After that, the convergence of the CSCD operators is discussed over Lipschitz-type space and in terms of modulus of continuity. In the next section, the weighted approximation result is obtained. Lastly, some estimates on the A-Statistical convergence of these operators are established. Chapter 5 provides the generalization of the family of Bernstein polynomials over a differ ent set of operators proposed by Mache and Zhou [66]. We investigate certain approximation properties for these operators, such as the rate of convergence via second-order modulus of continuity, Lipschitz space, Ditzian-Totik moduli of smoothness, Voronovskaya theorem, Gruss Voronovskaya theorem, and weighted approximation properties. Finally, we have illustrated the convergence of our operators graphically. In the next section of this chapter, we consider the Durrmeyer variant of modified Bernstein polynomials. First, we provide the auxiliary results and demonstrate the Bohman-Korovkin theorem. Then, we explore some approximation prop erties such as the rate of convergence using the Ditzian-Totik modulus of continuity, Vorono vaskaja type and weighted approximation theorem for these operators. Finally, the convergence behavior have been shown graphically. xi The aim of Chapter 6 is to introduce and study a new sequence of operators using Appell polynomials of class A 2 . First, the moments for these operators are established. Then, we study an estimate of error in approximation in terms of modulus of continuity and rate of convergence in weighted space for these operators. Finally, we obtain the rate of convergence for the function having the derivatives of bounded variation. The thesis is summarised in Chapter 7, before providing some insight into the author’s thoughts about the future research.