A Fractal Operator on Some Standard Spaces of Functions

Abstract

Through appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us 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 an α-fractal operator on , 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 paper we continue to investigate the fractal operator in more general spaces such as the space of all bounded functions and the Lebesgue space , and in some standard spaces of smooth functions such as the space of k-times continuously differentiable functions, Hölder spaces and Sobolev spaces . Using properties of the α-fractal operator, the existence of Schauder bases consisting of self-referential functions for these function spaces is established.