Study of Approximation Properties of Mixed Summation and Integral Type Operators

Abstract

newline This thesis deals with the study of approximation properties of summation and integral type positive linear operators. The content of the work is divided into 9 chapters. Chapter 1 presents basic concepts, definitions and historical background. Chapter 2 introduces Baskakov Durrmeyer type operators. We consider the class of the entire Lebesgue measurable functions and obtain central moments, Voronovskaja type expansion and error estimate in simultaneous approximation. Chapter 3 introduces Kantorovich type generalized Bernstein polynomial. We extend the results due to Lorentz and Voronowskaja for Lebesgue integrable functions. Chapter 4 introduces Kantorovich type generalized Baskakov operators. We obtain asymptotic expansion and error estimation in terms of second order of the modulus of continuityusing Steklov mean.We also obtain the rate of convergence for the functions with derivative of bounded variation. Chapter 5 studies Szasz Mirakyan Baskakov stancu kind operators. We obtain moments using hypergeometric seires, point wise convergence , asymptotic expansion and order of approximation in terms of higher order modulus of continuity using Steklov mean in simultaneous approximation. Chapter 6 Studies the convergence properties of modified positive linear operators introduced by Srivastava and Gupta in simultaneous approximation. Direct theorems which include point wise convergence, asymptotic expansion and error estimation in term of the modulus of continuity are acquired. Chapter 7 construct a new sequence of Beta Szasz type operators to make it to preserve linear functions using King approach. We derive point wise convergence using the Korovkin type theorem and an error estimate in terms of first order modulus of continuity then we claim a superior rate of convergence. Chapter 8 studies Lupas Beta operators and obtain a quantitative asymptotic formula and direct estimates in terms of Ditzian Totik modulus of continuity.