On Sequences of J. P. King-Type Operators

Tuncer Acar,Pedro Garrancho,Vita Leonessa,Mirella Cappelletti Montano

doi:10.1155/2019/2329060

Abstract

This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions andx2on[0,1]. Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend King’s approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the Szász-Mirakyan operators.

Highlights

The aim of this paper is to provide a survey on a series of recent investigations which are centered around the problem of obtaining better properties by modifying properly some well known sequences of positive linear operators in the underlying Banach function spaces
The paper is organized as follows: after a brief history on what has been done in this research area up to now, in Sections 3 and 4 we illustrate certain King-type modifications of the well-known Bernstein and Szasz-Mirakyan operators
From King’s work to nowadays, several investigations have been devoted to sequences of positive linear operators fixing certain functions

Summary

Introduction

The aim of this paper is to provide a survey on a series of recent investigations which are centered around the problem of obtaining better properties by modifying properly some well known sequences of positive linear operators in the underlying Banach function spaces. Such results are principally inspired by the pioneering work [1]. Through a quantitative estimate in terms of the Journal of Function Spaces classical first-order modulus of continuity, King proves that the order of approximation of Vn,rn∗ (f; x) to f(x) is at least as good as the order of approximation of Bn(f; x) to f(x) for 0 ≤ x < 1/3. The paper is organized as follows: after a brief history on what has been done in this research area up to now, in Sections 3 and 4 we illustrate certain King-type modifications of the well-known Bernstein and Szasz-Mirakyan operators

A Brief History

On Bernstein-Type Operators

On Szász-Mirakyan Type Operators