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ISSN 1088-6826(e) ISSN 0002-9939(p)

Weierstrass functions in Zygmund's class

Author(s): Yanick Heurteaux
Journal: Proc. Amer. Math. Soc. 133 (2005), 2711-2720.
MSC (2000): Primary 26A27, 28A80
Posted: March 22, 2005
MathSciNet review: 2146218
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Abstract | References | Similar articles | Additional information

Abstract: Consider the function

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

where

and

is an almost periodic $C^{1,\varepsilon}$ function. It is well known that the function

lives in the so-called Zygmund class. We prove that

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

which ensures that

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: 10.1090/S0002-9939-05-07857-3
PII: S 0002-9939(05)07857-3
Keywords: Weierstrass functions, almost periodic functions, second-order oscillations, Zygmund class
Received by editor(s): April 29, 2004
Posted: March 22, 2005
Communicated by: David Preiss
Copyright of article: Copyright 2005, American Mathematical Society

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