Approximation of Rough Functions

Abstract

For given $p\in [1,\infty ]$ 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 $\left|a\right|\ne {\left|b\right|}^{1/p}$. 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.

16 pages, 3 figures