Journal of Fourier Analysis and Applications

, Volume 16, Issue 1, pp 17–33

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


DOI: 10.1007/s00041-009-9072-2

Cite this article as:
Johnsen, J. J Fourier Anal Appl (2010) 16: 17. doi:10.1007/s00041-009-9072-2


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.


Nowhere-differentiabilityWeierstrass functionLacunary Fourier seriesSecond microlocalisation

Mathematics Subject Classification (2000)


Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborg ØstDenmark