Approximation Properties of Phillips Operators

Abstract

This paper deals with presenting a survey-cum-expository account of some developments concerning the approximation properties of well known Phillips operators. These operators are sometimes called as genuine Szasz Durrmeyer operators, because of their property of reproducing constants as well as linear functions. We give the alternate form to present these operators in terms of Hypergeometric functions, which are related to the modified Bessel’s function of first kind of index 1. Also, we observe that the r-th moment can be represented in terms of confluent hypergeometric functions, and further it can be written in terms of generalized Laguerre polynomials. In addition, we will present some known results on such operators, which include simultaneous approximation, linear and iterative combinations, global direct and inverse results, rate of convergence for functions of bounded variation, and q-analogues of these operators.