Approximation by Quantum Meyer K\"onig and Zeller-Fractal Functions

TL;DR

This paper introduces quantum fractal functions based on the Meyer-K"onig-Zeller operator, demonstrating their approximation capabilities, shape dependence, and fractal properties in various function spaces.

Contribution

The paper develops a new class of quantum MKZ fractal functions, analyzing their approximation properties, shape preservation, and fractal dimensions, extending classical results to quantum and fractal contexts.

Findings

Quantum MKZ fractal functions converge uniformly to continuous functions.

Existence of fractal functions that preserve order and bounds.

Analysis of fractal dimensions and approximation in $L^p$ spaces.

Abstract

In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-K\"onig-Zeller operator $M_{q,n}$. These quantum Meyer-K\"onig-Zeller (MKZ) fractal functions employ $M_{q,n} f$ as the base function in the iterated function system for $\alpha$-fractal functions. For $f\in C(I)$, $I$ closed in $\mathbb{R}$, it is shown that there exists a sequence of quantum MKZ fractal functions $\{f^{(q_n,\alpha)}_n\}_{n=0}^{\infty}$ which converges uniformly to $f$ without altering the scaling function $\alpha$. The shape of $f^{(q_n,\alpha)}_n$ depends on $q$ as well as the other IFS parameters. For $f,g\in C(I)$ with $g > 0$ or $f\geq g$, we show that there exists a sequence $\{f^{(q_n,\alpha)}_n\}_{n=0}^{\infty}$ with $f^{(q_n,\alpha)}_n \geq g$ converging to $f$. Quantum MKZ fractal versions of some classical M\"untz theorems are also presented. For $q=1$, the box dimension and some approximation-theoretic results of MKZ $\alpha$-fractal function are investigated in $C(I)$. Finally, MKZ $\alpha$-fractal functions are studied in $L^p$ spaces with ${p \geq 1}$.