Weierstrass functions in Zygmund's class
Résumé
Consider the function $$f(x)=\sum_{n=0}^{+\infty}b^{-n}g(b^nx)$$ where $b>1$ and $g$ is an almost periodic $C^{1,\eps}$ 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.