Simple proofs of nowhere-differentiability for Weierstrass's function and cases of slow growth

TL;DR

This paper presents a concise proof of the nowhere-differentiability of Weierstrass functions using basic integration theory and Fourier analysis, introducing a second microlocalisation technique to establish general results and explore functions with various frequency growth rates.

Contribution

It introduces a novel, simplified proof method for nowhere-differentiability and extends the theory to functions with polynomial and near-quadratic frequency growth.

Findings

Proof of nowhere-differentiability using basic integration theory

Development of a second microlocalisation technique in Fourier analysis

Examples with polynomial and almost quadratic frequency growth

Abstract

Using a few basics from integration theory, a short proof of nowhere-differentiability of Weierstrass functions is given. Restated in terms of the Fourier transformation, the method consists in principle of a second microlocalisation, which is used to derive two general results on existence of nowhere differentiable functions. Examples are given in which the frequencies are of polynomial growth and of almost quadratic growth as a borderline case.