DEGREE OF APPROXIMATION BY CERTAIN POSITIVE LINEAR OPERATORS

Abstract

The present thesis comprises of the convergence estimates of functions by several positive linear operators and their q-analogues. The thesis is divided into seven chapters. The Chapter 1 includes the literature survey, basic results and de nitions of approximation methods which will be required throughout the thesis. In the second chapter, we de ne a q-analogue of the modi ed Bernstein-Kantorovich operators introduced by Ozarslan and Duman (Numer. Funct. Anal. Optim., 37: 92-105, (2016)). We establish the shape preserving properties of these operators, e.g., convexity and monotonicity and study the approximation behaviour by virtue of Peetre's K-functional and Steklov mean. Further, we propose a bivariate generalization of these operators and obtain convergence estimates via partial and total modulus of continuity and K-functional. Finally, we introduce the corresponding GBS operators and investigate the convergence behaviour of the B 􀀀 continuous and B 􀀀 differentiable functions by virtue of mixed modulus of continuity. In the third chapter, we propose a bivariate generalization of modi ed -Bernstein operators. We establish a Voronovskaya type and Gr uss Voronovskaya type asymptotic theorems. Then, we obtain some approximation results using complete and partial moduli of continuity and compute an estimate of the error in terms of Peetre's K-functional. Further, we extend our study to the corresponding GBS operators and establish approximation results for functions in the B ogel space and the Lipschitz class of B 􀀀 continuous functions. The aim of the fourth chapter is to construct a sequence of the Bernstein- Durrmeyer type operators on a triangle and obtain approximation results by virtue of modulus of continuity and K-functional. Further, We construct a qth order generalization by means of a Taylor polynomial to approximate smooth functions. Lastly, the sequence of corresponding GBS operators is constructed and various approximation results are obtained. In the fth chapter, we de ne a q-analogue of the modi ed -Bernstein operators introduced by Kajla and Acar(Ann. Funct. Anal. 10(4): 570-582 (2019)). First, we give shape preserving properties of modi ed -Bernstein operators and then establish Voronovskaja type theorem. Also, we demonstrate local and global approximation theorems for these operators. Further, we propose a bivariate extension of these operators and examined convergence estimates using partial and total modulus of continuity and also extend the study to the corresponding GBS operators. In the sixth chapter, we construct a bivariate generalization by established a link between the Bernstein-Chlodowsky and the Sz asz-Appell-Kantorovich type operators. We study convergence properties by virtue of the total and partial modulus of continuity. Next, we determine convergence behaviour of these operator in a weighted space. Lastly, we construct the corresponding GBS operators and examined various approximation results for the B 􀀀 continuous and B 􀀀 differentiable functions. Chapter seven deals with the investigation of the approximation degree of Gupta- P alt aneaa operators introduced by Gupta (RACSAM, 113: 3717-3725, (2019)) by virtue of K-functional, Steklov mean and Voronovskaya type theorem. Some approximation results concerning the weighted approximation are also discussed. Lastly, approximation behaviour with the help of functions whose derivatives are of bounded variations are also considered.