Approximation of Rough Functions

For given $p\in [1,\mathrm{\infty }]$ 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.