SOME PROBLEMS IN APPROXIMATION FOR CERTAIN DISCRETE AND INTEGRAL OPERATORS

Abstract

In this thesis, the problems that we study are with respect to the approximation and error estimation of the linear positive operators. The techniques of simultaneous approximation and King type modi cation have been applied successfully to improve the order of approximation for various operators. Firstly, some theorems on approximation of the r-th derivative of a given function f by corresponding r-th derivative of the Durrmeyer variant of generalized Bernstein operator have been studied by contracting the interval of the de nition of integrability of function from class [0; 1] to [0; 1 􀀀 1 n+1]. The basic properties and Voronoskaya type results for the ordinary approximation for modi ed Baskakov operators and Bal azs operators have been studied and the results for better error estimation after considering King type modi cation of these operators have been obtained. Some results have been calculated for multidimensional Bernstein operators and its Durrmeyer variant. Quantitative global estimates for generalized double Baskakov operators have been studied. In the sequel, direct and inverse theorems for Beta Durrmeyer operators have been obtained. In the end, some approximation properties of modi ed Beta operators and an operator introduced by Jain with the help of Poisson type distribution have been studied, which include rate of convergence and statistical convergence.