APPROXIMATION BY COMBINATIONS OF OPERATORS OF SUMMATION-INTEGRAL TYPE

Abstract

In this thesis the problems that we study are with respect to the uniform norm and LP— norm. The techniques of linear and iterative combinations of positive linear operators has been applied successfully to improve the order of approximation for various operators and the central approximation theorems have been established for these combinations. Firstly, the inverse and saturation results are obtained for the linear combination of Szasz-Beta operators in Lp — norm. The iterative combinations of well known Bernstein-Durrmeyer and a new variant of Bernstein-Durrmeyer operators have been considered. The results are obtained in LP— norm for the iterates of Bernstein-Durrmeyer operators while for the iterates of other variant of Bernstein-Durrmeyer operators, the results are established in Tchebychev norm. The error estimates for sufficiently smooth functions in terms of the higher order of smoothness for Baskakov-Szasz and Phillips operators have been obtained. In the sequel for Bezier variant of certain summation-integral type operators, the direct and inverse theorems have been studied in terms of the Ditzian-Totik modulus of smoothness. In the end error estimates for certain summation-integral type operators for the functions having derivatives of bounded variation are obtained