Rate of convergence by Kantorovich-Sz\xe1sz type operators based on Brenke type polynomials

Purshottam Narain Agrawal,Serkan Araci,Tarul Garg

doi:10.1186/s13660-017-1430-z

Abstract

The present paper deals with the approximation properties of the univariate operators which are the generalization of the Kantorovich-Szász type operators involving Brenke type polynomials. We investigate the order of convergence by using Peetre’s K-functional and the Ditzian-Totik modulus of smoothness and study the degree of approximation of the univariate operators for continuous functions in a Lipschitz space, a Lipschitz type maximal function and a weighted space. The rate of approximation of functions having derivatives equivalent with a function of bounded variation is also obtained.

Highlights

Linear positive operators play an important role in the study of approximation theory
Taşdelen et al [ ] introduced a Kantorovich variant of the Szász operators based on Brenke type polynomials and discussed the order of convergence with the aid of the moduli of continuity and Peetre’s K
Mursaleen and Ansari [ ] presented a Chlodowsky type generalization of Szász operators defined by using Brenke type polynomials and studied the order of convergence for functions in a weighted space besides other classical approximation results

Summary

Introduction

Linear positive operators play an important role in the study of approximation theory. Öksüzer et al [ ] estimated the rate of convergence for functions of bounded variation for these operators by means of some results of probability theory. Aktaş et al [ ] considered a Stancu type modification of the Szász Kantorovich operators involving Brenke type polynomials and obtained the degree of approximation by means of the classical modulus of continuity and Peetre’s K-functional.

Results

Conclusion