ORDER OF CONVERGENCE BY CERTAIN POSITIVE LINEAR APPROXIMATION METHODS

Abstract

The thesis is an e ort in the constructive approximation by providing multifarious type of linear positive operators to approximate di erent classes of functions. It is branched into eight chapters. The rst chapter encompasses the literature and important de nitions. In Chapter 2, we discuss some approximation properties of a bi-variate Kantorovich-Stancu type operators by means of various approximation tools. We provide a generalization of the proposed bi-variate operator based on the Taylor's polynomials, and discuss some more such type of generalizations of well-known operators in the application section. From Chapter 3, we take our interest in the direction of the application of quantum/post-quantum calculus in approximation theory. The chapter three provides a q-analogue of the alpha-Baskakov operators. We study approximation behavior of the operators in di erent weighted spaces using weighted modulus of continuity and Peetre's K-functional. Further, we extended the study to the bi-variate and GBS cases. In Chapter 4, we rst de ne (somewhat a modi ed de nition) a linear positive operator based on the multivariate q-Lagrange polynomials and then extended the operator's de nition in bi-variate and GBS settings. We establish some approximation results in terms of the partial modulus of continuities and the complete modulus of continuity. A generalization of the proposed operator is also de ned in order to investigate the approximation behavior for su ciently smooth functions in the bi-variate settings. Summability theory can be considered as a sister branch of approximation theory, and so we have tried to connect the dots between them. In this regard, in Chapter 5, we study the convergence of an operator based on multivariate (p; q)- Lagrange polynomials by using the P-summability technique. The Chapter 6, is the continuation of the work done in Chapter 4 in which, using the q-Riemann integral, we introduce an integral type operator and studied its convergence through deferred weighted A-statistical convergence and P-summability methods. In Chapter 7, we provide two non-trivial Korovkin type theorems for any general sequence of linear positive operators in the settings of deferred weighted A-statistical convergence and P-summability methods. Furthermore, we provide an example of q-Lagrange-Hermite operator and brie y elaborate the applicability of our general theorems. Finally, we conclude the thesis in Chapter 8, with some comments and future scope.