On Some Extensions of Szasz Operators Including Boas‐Buck‐Type Polynomials

Sezgin Sucu,Gürhan İçöz,Serhan Varma

doi:10.1155/2012/680340

Abstract

This paper is concerned with a new sequence of linear positive operators which generalize Szasz operators including Boas‐Buck‐type polynomials. We establish a convergence theorem for these operators and give the quantitative estimation of the approximation process by using a classical approach and the second modulus of continuity. Some explicit examples of our operators involving Laguerre polynomials, Charlier polynomials, and Gould‐Hopper polynomials are given. Moreover, a Voronovskaya‐type result is obtained for the operators containing Gould‐Hopper polynomials.

Highlights

The approximation theory, which is concerned with the approximation of functions by simpler calculated functions, is a branch of mathematical analysis
For g z 1, in view of the generating functions 1.2, we find pk x xk/k! and from 1.3 we meet again the Szasz operators given by 1.1
Ismail 4 obtained another generalization of the Szasz operators 1.1 and Jakimovski and Leviatan operators 1.3 by means of Sheffer polynomials

Summary

Introduction

The approximation theory, which is concerned with the approximation of functions by simpler calculated functions, is a branch of mathematical analysis. Ismail 4 obtained another generalization of the Szasz operators 1.1 and Jakimovski and Leviatan operators 1.3 by means of Sheffer polynomials. Varma et al 5 constructed linear positive operators including Brenke-type polynomials. Brenke-type polynomials 6 have generating functions of the form. Ii B : 0, ∞ −→ 0, ∞ , 1.10 iii 1.7 and the power series 1.8 converge for |t| < R R > 1 , Varma et al introduced the following linear positive operators involving the Brenke-type polynomials. Our aim is to construct linear positive operators by using Boas-Bucktype polynomials including the Brenke-type polynomials, Sheffer polynomials, and Appell polynomials with special cases. Given the above restrictions, we present a new form of linear positive operators with Boas-Buck-type polynomials as follows: Bn f ; x :

A 1 B nxH 1

Approximation Properties of Bn Operators

H 1 A 1 B nxH 1 nA 1 B nxH 1 x

Examples