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Weierstrass functions in Zygmund's class


Author: Yanick Heurteaux
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
Published electronically: March 22, 2005
MathSciNet review: 2146218
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Abstract: Consider the function

\begin{displaymath}f(x)=\sum_{n=0}^{+\infty}b^{-n}g(b^nx)\end{displaymath}

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

DOI: https://doi.org/10.1090/S0002-9939-05-07857-3
Keywords: Weierstrass functions, almost periodic functions, second-order oscillations, Zygmund class
Received by editor(s): April 29, 2004
Published electronically: March 22, 2005
Communicated by: David Preiss
Article copyright: © Copyright 2005 American Mathematical Society

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