A STUDY ON ESTIMATES OF CONVERGENCE OF CERTAIN APPROXIMATION OPERATORS

Abstract

This thesis is mainly a study of convergence estimates of various approximation opera tors. Approximation theory is indeed an old topic in mathematical analysis that remains an appealing field of study with several applications. The findings presented here are related to the approximation of specific classes of linear positive operators. The introduc tory chapter is a collection of relevant definitions and literature of concepts that are used throughout this thesis. The second chapter is based on approximation of certain exponential type opera tors. The first section of this chapter presents the study of convergence estimates of Kan torovich variant of Ismail-May operators. Further, a two variable generalisation of the proposed operators is also discussed. The second section is dedicated to a modification of Ismail-May exponential type operators which preserve functions of exponential growth. The modified operators in general are not of exponential type. In chapter three, we present a Durrmeyer type construction involving a class of or thogonal polynomials called Apostol-Genocchi polynomials and Palt ˇ anea operators with ˇ real parameters α, λ and ρ. We establish approximation estimates such as a global approx imation theorem and rate of approximation in terms of usual, r−th and weighted modulus of continuity. We further study asymptotic formulae such as Voronovskaya theorem and quantitative Voronovskaya theorem. The rate of convergence of the proposed operators for the functions whose derivatives are of bounded variation is also presented. Inspired by the King’s approach, chapter four deals with the preservation of func tions of the form t s , s ∈ N ∪ {0}. Followed by some useful lemmas, we determine the rate of convergence of the proposed operators in terms of usual modulus of continuity and Peetre’s K- functional. Further, the degree of approximation is also established for the function of bounded variation. We also illustrate via figures and tables that the proposed modification provides better approximation for preservation of test function e3. In chapter five, we consider a Kantorovich variant of the operators proposed by Gupta and Holhos (68) using arbitrary sequences which preserves the exponential func tions of the form a −x . It is shown that the order of approximation can be made better xi xii Abstract with appropriate choice of sequences with certain conditions. We therefore provide nec essary moments and central moments and some useful lemmas. Further, we present a quantitative asymptotic formula and estimate the error in approximation. Graphical rep resentations are provided in the end with different choices of sequences satisfying the given conditions. The last chapter summarizes the thesis with a brief conclusion and also discusses the future prospects of this thesis.