Approximation by Quantum Meyer-König-Zeller Fractal Functions

Abstract

In this paper, a novel class of quantum fractal functions is introduced based on the Meyer-König-Zeller operator Mq,n. These quantum Meyer-König-Zeller (MKZ) fractal functions employ Mq,nf as the base function in the iterated function system for α-fractal functions. For f∈C(I), I closed interval in R, it is shown that a sequence of quantum MKZ fractal functions {fn(qn,α)}n=0∞ exists which converges uniformly to f without altering the scaling function α. The shape of fn(qn,α) depends on q as well as the other iterated function system parameters. For f,g∈C(I), f≥g>0, we show that a sequence {fn(qn,α)}n=0∞ exists with fn(qn,α)≥g>0 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 functions are investigated in C(I). Finally, MKZ α-fractal functions are studied in Lp spaces with p≥1.