Approximation of Rough Functions

TL;DR

This paper investigates solutions to a functional equation involving rough functions, establishing existence and uniqueness in L^p spaces, and explores implications for fractal interpolation, approximation theory, and Fourier analysis.

Contribution

It provides the first comprehensive analysis of solutions to the equation for rough functions, including classical nowhere differentiable functions, with applications to multiple mathematical fields.

Findings

Existence and uniqueness of solutions in L^p spaces.

Solutions include classical nowhere differentiable functions.

Connections to fractal interpolation and Fourier analysis.

Abstract

For given $p\in\lbrack1,\infty]$ and $g\in L^{p}\mathbb{(R)}$, we establish the existence and uniqueness of solutions $f\in L^{p}(\mathbb{R)}$, to the equation \[ f(x)-af(bx)=g(x), \] where $a\in\mathbb{R}$, $b\in\mathbb{R} \setminus \{0\}$, and $\left\vert a\right\vert \neq\left\vert b\right\vert ^{1/p}$. Solutions include well-known nowhere differentiable functions such as those of Bolzano, Weierstrass, Hardy, and many others. Connections and consequences in the theory of fractal interpolation, approximation theory, and Fourier analysis are established.