Simple Proofs of Nowhere-Differentiability for Weierstrass’s Function and Cases of Slow Growth

摘要

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. Keywords Nowhere-differentiability - Weierstrass function - Lacunary Fourier series - Second microlocalisation Mathematics Subject Classification (2000) 26A27 Communicated by David Walnut.