APPROXIMATION OF FUNCTIONS BY CERTAIN POSITIVE LINEAR METHODS OF CONVERGENCE

Abstract

Approximation theory is indeed an old topic in mathematical analysis that continues to be an interesting field of research with several applications. After the well-known theorem due to Weierstrass and the important convergence theorem of Korovkin, many new opera tors were proposed and constructed by several researchers. The theory of these operators has been an important area of research in the last few decades. This thesis is mainly concerned with convergence estimates of several positive linear operators. The introduc tory chapter is a collection of relevant definitions and literature of concepts that are used throughout this thesis. Several results have been established for different exponential-type operators, a no tion that was first presented by May and then extensively investigated in cooperation with Ismail. The Bézier variant of these operators has been defined. Two decades ago it was observed that if we modify the original operators, we can have a better approximation. The basic properties and Voronoskaya type results for the approximation of exponential operators have been studied, and after being modified to preserve exponential functions, the results for improved error estimates have been achieved. A modification of certain Gamma type operators that preserves the test functions t ϑ , ϑ = {0} S N has been pro vided and rate of convergence for functions of bounded variation has been studied. Some approximation properties of the Pólya distribution-based generalization of λ Bernstein operators, such as rate of convergence, interpolation behavior, and the impact of changing parameter values, have been investigated. Certain theorems are derived to verify the convergence of generalized Bernstein operators based on shifted knots. Some results have been proved for bivariate generalization of operators involving a class of orthogonal polynomials called Apostol-Genocchi polynomials. Furthermore, a conceptual extension for these bivariate operators, referred to as the "generalized boolean sum (GBS)", has been introduced with the goal of determining the degree of approxima tion for Bögel continuous functions. Graphical illustration and tables that effectively showcase the convergence and demonstrate the approximation error have been included for all the operators.