On the Approximation by Balázs–Szabados Operators

Adrian Holhoş

doi:10.3390/math9141588

Abstract

We present three new approximation properties of the Balázs–Szabados operators. Firstly, we prove that, in certain cases, these operators approximate some super-exponential functions on compact intervals. Next, we provide a new estimate of the error of approximation using a suitable modulus of continuity. Finally, we characterize the functions which can be uniformly approximated in the weighted norm of polynomial weight spaces.

Highlights

The problem of Weierstrass [1,2] to approximate continuous functions using a sequence of polynomials initiated this research field, but the systematic study of these operators began only in the 1960s with the books of Korovkin [3,4] and Lorentz [5]
Korovkin has discovered simple conditions to verify if a sequence of positive linear operators forms an approximation process, and since there have been constructed many operators
The growth can be enlarged indefinitely, in some sense, and this is correlated with the proper choice of the parameter β. Another interesting thing about the Balázs–Szabados operators is that they are built using only a finite sum and not a series and, so, the possibility of approximating functions with such a high growth comes as a surprise

Summary

Introduction

The study of positive linear operators is an important research area in approximation theory. Another interesting thing about the Balázs–Szabados operators is that they are built using only a finite sum and not a series and, so, the possibility of approximating functions with such a high growth comes as a surprise This result is presented in Theorem 1 of Section 2. By considering a generalization of the operators (1), Agratini ([39], Theorem 5) has given a negative result by presenting an example of a function which cannot be approximated in the weighted norm. He has estimated the error of approximation, too, but only pointwise. We will denote by Cw ( I ) the space of functions from Bw ( I ) which are continuous on I

Approximation on Compact Intervals of Super-Exponential Functions

Estimation of the Rate of Uniform Approximation

Weighted Approximation in Polynomial Weight Spaces