A Fractal Operator on Some Standard Spaces of Functions

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.