Higher Order Approximation of Continuous Functions by a Modified Meyer-K\"{o}nig and Zeller-Type Operator

TL;DR

This paper introduces a modified Meyer-König and Zeller operator for better approximation of continuous functions on [0,1), providing error estimates and proving direct and converse theorems, with improved approximation order over previous variants.

Contribution

It presents a new Goodman-Sharma type modification of the Meyer-König and Zeller operator, achieving superior approximation order despite not being positive.

Findings

The operator provides a better order of approximation than previous variants.

Error estimates are established in relation to a K-functional.

The operator is linear but not positive, yet more effective in approximation.

Abstract

A new Goodman-Sharma type modification of the Meyer-K\"{o}nig and Zeller operator for approximation of bounded continuous functions on [0,1) is presented. We estimate the approximation error of the proposed operator and prove direct and strong converse theorems with respect to a related K-functional. The operator is linear but not a positive one. However it benefits a better order of approximation compared to the Goodman-Sharma variant of Meyer-K\"{o}nig and Zeller type operator investigated by Ivanov and Parvanov in 2012.