On Better Approximation Order for the Max-Product Meyer-K\"onig and Zeller Operator

TL;DR

This paper improves the approximation order of the max-product Meyer-König and Zeller operator, showing it can be better than previously claimed without restricting the function class, by deriving a new degree of approximation.

Contribution

The authors demonstrate a better approximation order for the operator, surpassing earlier limitations, using classical modulus of continuity without narrowing the function class.

Findings

New approximation order: (1-y) y^{1/α} / m^{1 - 1/α}

Improvement over previous order by Bede et al.

Approximation order approaches 1 as α increases.

Abstract

In [5], Bede et al. defined the max-product Meyer-K\"{o}nig and Zeller operator. They examined the approximation and shape preserving properties of this operator, and they found the order of approximation to be $\frac{\sqrt{y}\left( 1-y\right) }{\sqrt{m}}$ by the modulus of continuity and claimed that this order of approximation could only be improved in certain subclasses of the functions. In contrast to this claim, we demonstrate that we can obtain a better order of approximation without reducing the function class by the classical modulus of continuity. We find the degree of approximation to be $\frac{\left( 1-y\right) y^{\frac{1}{% \alpha }}}{m^{1-\frac{1}{\alpha }}}$, $\alpha =2,3,...$ . Since $1-\frac{1}{% \alpha }$ tends to $1$ for enough big $\alpha $, we improve this degree of approximation.