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

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 ${p \geq 1}$.