RATE OF CONVERGENCE OF CERTAIN POSITIVE LINEAR APPROXIMATION METHODS

Abstract

In the present thesis, the focus is primarily on the study of approximation properties of some well known linear positive operators and their new generalizations e.g. the Stancu type generalization, bivariate extension, B´ezier variant and q−variant of the well known operators. We divide the thesis into eight chapters. The chapter 0 is of introductory nature it includes the literature survey, basic definitions and notations of approximation methods which will be used throughout the thesis. In chapter 1, we obtain the rate of convergence in ordinary and simultaneous approximation, statistical convergence for generalized Baskakov-Sz´asz type operators. Also, we estimate the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation. In chapter 2, we study some approximation properties of the Sz´asz type operators involving Charlier polynomials and their Kantorovich modification. We establish the direct results in a Lipschitz type space, weighted approximation theorems and the rate of approximation of functions having derivatives of bounded variation for both of these operators. We also obtain A−statistical convergence for the Kantorovich case. The third chapter deals with the Durrmeyer type modification of the Sz´asz type operators based on Charlier polynomials and establish a Voronovskaja type asymptotic result, local direct results, weighted approximation, statistical convergence and approximation of functions with derivatives of bounded variation for these operators have been discussed. In chapter 4, we consider the Kantorovich modification of the Lupa¸s operators based on Polya distribution. The Voronovskaja type asymptotic formula, local and global approximation results and the rate of convergence for absolutely continuous ifunctions having a derivative equivalent with a function of bounded variation have been discussed for these operators. Lastly, we introduce the bivariate extension of these operators and obtain the order of approximation using Peetre’s K−functional. We also show the convergence of these operators to certain functions by graphics in Matlab for both single and bivariate cases and also estimate the error in the approximation for the two dimensional case. The fifth chapter deals with the B´ezier variant of summation integral type operators having Polya and Bernstein basis functions and a direct approximation theorem with the aid of the Ditzian-Totik modulus of smoothness and also study the rate of convergence for absolutely continuous functions having a derivative equivalent with a function of bounded variation are established. The sixth chapter includes the bivariate extension of the Lupa¸s-Durrmeyer operators based on Polya distribution and obtain a Voronovskaja type theorem and the rate of convergence. Then, we introduce the Generalized Boolean Sum(GBS) of these operators and estimate the degree of approximation by means of the mixed modulus of smoothness. In the last chapter, we propose the q−analogue of the modified Baskakov-Sz´asz- Stancu operators. We obtain the moments of the operators and then study some direct results e.g. Voronovskaja type asymptotic theorem and the rate of convergence in terms of the weighted modulus of continuity. Further, we discuss the point-wise estimation using the Lipschitz type maximal function and the rate of A−statistical convergence of these operators by using weighted modulus of continuity