Weierstrass functions in Zygmund’s class

Abstract:

Consider the function \[ f(x)=\sum _{n=0}^{+\infty }b^{-n}g(b^nx)\] where $b>1$ and $g$ is an almost periodic $C^{1,\varepsilon }$ function. It is well known that the function $f$ lives in the so-called Zygmund class. We prove that $f$ is generically nowhere differentiable. This is the case in particular if the elementary condition $g^\prime (0)\not = 0$ is satisfied. We also give a sufficient condition on the Fourier coefficients of $g$ which ensures that $f$ is nowhere differentiable.