Local fractal functions and function spaces

TL;DR

This paper introduces local iterated function systems and local fractal functions, establishing their properties and conditions for inclusion in various classical function spaces, thereby advancing fractal analysis and function space theory.

Contribution

It presents a new class of local fractal functions derived from local iterated function systems and provides formulas for their inclusion in key function spaces.

Findings

Defined local fractal functions and their properties

Derived formulas for function space inclusion

Extended fractal analysis to classical function spaces

Abstract

We introduce local iterated function systems and present some of their basic properties. A new class of local attractors of local iterated function systems, namely local fractal functions, is constructed. We derive formulas so that these local fractal functions become elements of various function spaces, such as the Lebesgue spaces $L^p$, the smoothness spaces $C^n$, the homogeneous H\"older spaces $\dot{C}^s$, and the Sobolev spaces $W^{m,p}$.