Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Weierstrass functions in Zygmund's class

Author: Yanick Heurteaux
Journal: Proc. Amer. Math. Soc. 133 (2005), 2711-2720
MSC (2000): Primary 26A27, 28A80
Published electronically: March 22, 2005
MathSciNet review: 2146218
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider the function


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.

References [Enhancements On Off] (What's this?)

  • 1. T. Bousch and Y. Heurteaux, On oscillations of Weierstrass-type functions, manuscript, 1999.
  • 2. -, Caloric measure on domains bounded by Weierstrass-type graphs, Ann. Acad. Sci. Fenn. 25 (2000), 501-522.MR 1762432 (2001h:31004)
  • 3. C. Corduneanu, Almost periodic functions, Interscience, New York, 1968.MR 0481915 (58:2006)
  • 4. K. Falconer, Fractal Geometry : Mathematical Foundations and Applications, John Wiley & Sons Ltd., New-York, 1990. MR 1102677 (92j:28008)
  • 5. G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc. 17 (1916), 301-325. MR 1501044
  • 6. Y. Heurteaux, Weierstrass functions with random phases, Trans. Amer. Math. Soc. 355 (2003), 3065-3077. MR 1974675 (2004f:26012)
  • 7. T.-Y. Hu and K.-S. Lau, Fractal dimensions and singularities of the Weierstrass type functions, Trans. Amer. Math. Soc. 335 (1993), 649-665. MR 1076614 (93d:28011)
  • 8. J.L. Kaplan, J. Mallet-Paret, and J.A. Yorke, The Lyapunov dimension of a nowhere differentiable attracting torus, Ergod. Th. & Dynam. Sys. 4 (1984), 261-281. MR 0766105 (86h:58091)
  • 9. S. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Exposition. Math. 1 (1983), 193-260.MR 0782608 (86g:41001)
  • 10. R. D. Mauldin and S. C. Williams, On the Hausdorff dimension of some graphs, Trans. Amer. Math. Soc. 298 (1986), 793-804.MR 0860394 (88c:28006)
  • 11. F. Przytycki and M. Urbanski, On the Hausdorff dimension of some fractal sets, Studia Math. 93 (1989), 155-186.MR 1002918 (90f:28006)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 26A27, 28A80

Retrieve articles in all journals with MSC (2000): 26A27, 28A80

Additional Information

Yanick Heurteaux
Affiliation: Laboratoire de Mathématiques, UMR 6620, Université Blaise Pascal, F-63177 Aubière cedex, France

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

American Mathematical Society