Revista Matemática Iberoamericana

Hölder exponents of arbitrary functions

Antoine Ayache and Stéphane Jaffard

Full-text: Open access

Abstract

The functional class of Hölder exponents of continuous function has been completely characterized by P. Andersson, K. Daoudi, S. Jaffard, J. Lévy-Véhel and Y. Meyer [Andersson, P.: Wavelets and local regularity. PhD Thesis. Department of Mathematics, G\"oteborg, 1997], [Andersson, P.: Characterization of pointwise Hölder regularity. Appl. Comput. Harmon. Anal. {\bf 4} (1997), 429-443], [Daoudi, K., Lévy-Véhel J. and Meyer, Y.: Construction of continuous functions with prescribed local regularity. Constr. Approx. {\bf 14} (1998), 349-385], [Jaffard, S.: Functions with prescribed Hölder exponent. Appl. Comput. Harmon. Anal. {\bf 2} (1995), 400-401]; these authors have shown that this class exactly corresponds to that of the lower limits of the sequences of nonnegative continuous functions. The problem of determining whether or not the Hölder exponents of discontinuous (and even unbounded) functions can belong to a larger class remained open during the last decade. The main goal of our article is to show that this is not the case: the latter Hölder exponents can also be expressed as lower limits of sequences of continuous functions. Our proof mainly relies on a ``wavelet-leader'' reformulation of a nice characterization of pointwise Hölder regularity due to P. Anderson.

Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 1 (2010), 77-89.

Dates
First available in Project Euclid: 16 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1266330117

Mathematical Reviews number (MathSciNet)
MR2666308

Zentralblatt MATH identifier
1203.26021

Subjects
Primary: 26B35: Special properties of functions of several variables, Hölder conditions, etc. 42C40: Wavelets and other special systems 65T60: Wavelets

Keywords
Hölder regularity Hölder exponents wavelets

Citation

Ayache, Antoine; Jaffard, Stéphane. Hölder exponents of arbitrary functions. Rev. Mat. Iberoamericana 26 (2010), no. 1, 77--89. https://projecteuclid.org/euclid.rmi/1266330117


Export citation

References

  • Andersson, P.: Wavelets and local regularity. PhD Thesis. Department of Mathematics, Göteborg, 1997.
  • Andersson, P.: Characterization of pointwise Hölder regularity. Appl. Comput. Harmon. Anal. 4 (1997), 429-443.
  • Ayache, A., Jaffard S. and Taqqu, M.: Wavelet construction of generalized multifractional processes. Rev. Mat. Iberoamericana 23 (2007), no. 1, 327-370.
  • Bruckner, A.: Differentiation of real functions. Second edition. CRM Monograph Series 5. American Mathematical Society, Providence, RI, 1994.
  • Calderón A.P. and Zygmund, A.: Local properties of solutions of elliptic partial differential equations. Studia Math. 20 (1961), 171-225.
  • Daoudi, K., Lévy-Véhel J. and Meyer, Y.: Construction of continuous functions with prescribed local regularity. Constr. Approx. 14 (1998), 349-385.
  • Guikhman, I. and Skorokhod, A.: Introduction à la théorie des processus aléatoires. Mir Pub. (1977), french translation (1980).
  • Jaffard, S.: Construction de fonctions multifractales ayant un spectre de singularités prescrit. C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), 19-24.
  • Jaffard, S.: Functions with prescribed Hölder exponent. Appl. Comput. Harmon. Anal. 2 (1995), 400-401.
  • Jaffard, S.: Wavelet techniques in multifractal analysis. In Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2, 91-152. Proc. Sympos. Pure Math. 72, Part 2. Amer. Math. Soc., Providence, RI, 2004.
  • Jaffard, S.: Wavelet techniques for pointwise regularity. Ann. Fac. Sci. Toulouse Math. (6) 15 (2006), 3-33.
  • Lemarié, P.-G. and Meyer, Y.: Ondelettes et bases hilbertiennes. Rev. Mat. Iberoamericana 2 (1986), 1-18.
  • Lévy-Véhel, J. and Seuret, S.: The local Hölder function of a continuous function. Appl. Comput. Harmon. Anal. 13 (2002), 263-276.
  • Meyer, Y.: Wavelets, vibrations and scalings. CRM Monograph Series 9. American Mathematical Society, Providence, RI, 1998.
  • Rudin, W.: Real and complex analysis. Mcgraw-Hill International Editions, Mathematics series, 1986.