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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Weierstrass functions in Zygmund’s class
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by Yanick Heurteaux PDF
Proc. Amer. Math. Soc. 133 (2005), 2711-2720 Request permission

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.
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Additional Information
  • Yanick Heurteaux
  • Affiliation: Laboratoire de Mathématiques, UMR 6620, Université Blaise Pascal, F-63177 Aubière cedex, France
  • Email: Yanick.Heurteaux@math.univ-bpclermont.fr
  • Received by editor(s): April 29, 2004
  • Published electronically: March 22, 2005
  • Communicated by: David Preiss
  • © Copyright 2005 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2711-2720
  • MSC (2000): Primary 26A27, 28A80
  • DOI: https://doi.org/10.1090/S0002-9939-05-07857-3
  • MathSciNet review: 2146218