CERTAIN APPROXIMATION METHODS OF CONVERGENCE FOR LINEAR POSITIVE OPERATORS

Abstract

The present thesis deals with the approximation behavior of various linear positive operators, their modifications, and approximation properties like rate of convergence, error estimation, and graphical comparison. We segregate the thesis in seven chapters. Chapter 1 is an introduction that contains a history of approximation theory, basic definitions, and converging tools which play an important role in approximation theory. Chapter 2 is divided into three sections. In the first section, we have considered the Kantorovich form of α−Bernstein operators introduced by Chen et al. [36]. We discussed some auxiliary properties and study the direct local approximation theorem, Voronovskaya type asymptotic, and function of bounded variation for α−BernsteinKantorovich operators. In the second section, we have considered the q-analogue of α-Bernstein Kantorovich operators. For these proposed operators, we studied some convergence properties by using first and second-order modulus of continuity. In the last section, we have proposed the Stancu type generalization of the family of Bernstein Kantorovich operators involving parameter α ∈ [0,1] with Shifted Knots. These operators provide the pliability to approximate on the interval [0,1] and over its subintervals. For the proposed operators, we investigate some basic results of approximation and their rate of convergence in terms of first and second-order modulus of continuity, Lipschitz class, and Lipschitz-type function. We also estimate the global rate of convergence of the operators with the help of the Ditzian-Totik modulus of smoothness. Moreover, the p th order generalization of the operators is established. Some numerical simulation and graphical comparisons are given for a better depiction of theoretical results. xiii In chapter 3, we propose the Kantorovich type generalized Szász-Mirakyan operators based on Jain and Pethe operators [101]. We study local approximation results in terms of classical modulus of continuity as well as Ditzian-Totik moduli of smoothness. Further, we establish the rate of convergence in a class of absolutely continuous functions having a derivative coinciding a.e. with a function of bounded variation. In chapter 4, we propose the integral form of Jain and Pethe operators associated with the Baskakov operators and study some basic properties. We estimate the rate of convergence, Voronovskaja type asymptotic estimate formula, and weighted approximation of these operators. In chapter 5, we consider new operators, which are defined by Gupta and Srivastava [90]. They considered a general sequence of positive linear operators and gave the modified form of their previous operators [142]. As these operators preserve linear functions, we call these operators as genuine Gupta-Srivastava operators. Here we discuss some basic properties, direct results, rate of convergence for a class of functions whose derivatives are of bounded variation, and weighted approximation for our considered operators. In chapter 6, we propose the Bézier variant of the Gupta-Srivastava operators [90] and discuss some direct convergence results by using Lipschitz type spaces, Ditzian-Totik modulus of smoothness, weighted modulus of continuity, and for functions whose derivatives are of bounded variation. In the end, some graphical representation for comparison with other variants has been presented. In chapter 7, we consider mixed approximation operators based on the second-kind beta transform by using Szász-Mirakyan operators. For the proposed operators, we establish direct result, Voronovskaya type theorem, quantitative Voronovskaya type theorem, Grüss Voronovskaya type theorem, weighted approximation, and functions of bounded variation.