Approximation by Quantum Meyer König and Zeller-Fractal Functions

Abstract

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 $\hat{I}\pm$-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 $\{f_n^{(q_n,\hat{I}\pm )}{\}}_{n=0}^{\infty }$ which converges uniformly to $f$ without altering the scaling function $\hat{I}\pm$. The shape of $f_n^{(q_n,\hat{I}\pm )}$ depends on $q$ as well as the other IFS parameters. For $f,g\in C(I)$ with $g>0$ or $f\ge g$, we show that there exists a sequence $\{f_n^{(q_n,\hat{I}\pm )}{\}}_{n=0}^{\infty }$ with $f_n^{(q_n,\hat{I}\pm )}\ge g$ 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 $\hat{I}\pm$-fractal function are investigated in $C(I)$. Finally, MKZ $\hat{I}\pm$-fractal functions are studied in $L^p$ spaces with $p\ge 1$.