A Fractal Operator on Some Standard Spaces of Functions

TL;DR

This paper extends the concept of the $ ext{α}$-fractal operator from continuous functions to various function spaces, establishing properties and the existence of Schauder bases of self-referential functions.

Contribution

It generalizes the $ ext{α}$-fractal operator to broader function spaces and proves the existence of Schauder bases of self-referential functions within these spaces.

Findings

The $ ext{α}$-fractal operator is well-defined on various function spaces.

Existence of Schauder bases of self-referential functions is established.

Connections between fractal functions and classical function spaces are demonstrated.

Abstract

By appropriate choices of elements in the underlying iterated function system, methodology of fractal interpolation entitles one to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as $\alpha$-fractal operator on $\mathcal{C}(I)$, the space of all real-valued continuous functions defined on a compact interval $I$. With an eye towards connecting fractal functions with other branches of mathematics, in this article, we continue to investigate fractal operator in more general spaces such as the space $\mathcal{B}(I)$ of all bounded functions and Lebesgue space $\mathcal{L}^p(I)$, and some standard spaces of smooth functions such as the space $\mathcal{C}^k(I)$ of $k$-times continuously differentiable functions, H\"{o}lder spaces $\mathcal{C}^{k,\sigma}(I)$, and Sobolev spaces $\mathcal{W}^{k,p}(I)$. Using properties of the $\alpha$-fractal operator, the existence of Schauder basis consisting of self-referential functions is established.