Linear Positive Operators Involving Orthogonal Polynomials

In recent years, there has been a significant increase in activities in approximation of continuous functions on the semi-real axis by the linear positive operators based on orthogonal polynomials. Jakimovski and Leviatan initiated the study in this direction by defining the generalization of Szasz operators based on Appell polynomials. Ismail generalized the Szasz operators by means of Sheffer polynomials. In the literature, several sequences of linear positive operators have been defined, involving orthogonal polynomials, and their Durrmeyer and Kantorovich-type variants have been investigated. This chapter aims to make a survey of the research work available on approximation by the linear positive operators defined by using orthogonal polynomials. Sidharth et al. introduced the Szasz-Durrmeyer-type polynomials based on Boas-Buck-type polynomials and established a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. The approximation of functions whose derivative are locally of bounded variation are also discussed.