arXiv2022
In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-König-Zeller operator $M_{q,n}$. These quantum Meyer-König-Zeller (MKZ) fractal functions employ $M_{q,n} f$ as the base function in the iterated function system for $α$-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 $\{f^{(q_n,α)}_n\}_{n=0}^{\infty}$ which converges uniformly to $f$ without altering the scaling function $α$. The shape of $f^{(q_n,α)}_n$ 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 $\{f^{(q_n,α)}_n\}_{n=0}^{\infty}$ with $f^{(q_n,α)}_n \geq g$ converging to $f$. Quantum MKZ fractal versions of some classical Müntz theorems are also presented. For $q=1$, the box dimension and some approximation-theoretic results of MKZ $α$-fractal function are investigated in $C(I)$. Finally, MKZ $α$-fractal functions are studied in $L^p$ spaces with ${p \geq 1}$.