Advertisement

Journal d'Analyse Mathématique

, Volume 124, Issue 1, pp 387–408 | Cite as

Multifractal analysis of the divergence of Fourier series: The extreme cases

  • Frédéric Bayart
  • Yanick Heurteaux
Article

Abstract

We study the size, in terms of the Hausdorff dimension, of the subsets of T such that the Fourier series of a generic function in L 1(T), L p (T), or C(T) may behave badly. Genericity is related to the Baire Category Theorem or the notion of prevalence. This paper is a continuation of [3].

Keywords

Fourier Series Lebesgue Measure Hausdorff Dimension Trigonometric Polynomial MULTIFRACTAL Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Arias de Reyna, Pointwise Convergence of Fourier Series, Lecture Notes in Mathematics 1785, Springer, Berlin, 2002.CrossRefzbMATHGoogle Scholar
  2. [2]
    J-M. Aubry, On the rate of pointwise divergence of Fourier and wavelet series in Lp, J. Approx. Theory 138 (2006), 97–111.CrossRefzbMATHMathSciNetGoogle Scholar
  3. [3]
    F. Bayart and Y. Heurteaux, Multifractal analysis of the divergence of Fourier series Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), 927–946.zbMATHMathSciNetGoogle Scholar
  4. [4]
    V. Beresnevich and S. Velani, A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures, Ann. of Math. (2) 164 (2006), 971–992.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255–260.CrossRefMathSciNetGoogle Scholar
  6. [6]
    K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nd ed., John Wiley & Sons, Inc. Hoboken, NJ, 2003.CrossRefGoogle Scholar
  7. [7]
    A. Fraysse and S. Jaffard, How smooth is almost every function in a Sobolev space?, Rev. Mat. Iberoam. 22 (2006), 663–682.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    A. Fraysse, S. Jaffard, and J. P. Kahane, Quelques propriétés génériques en analyse, C. R. Math. Acad. Sci. Paris 340 (2005), 645–651.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, Cambridge, 1995.CrossRefzbMATHGoogle Scholar
  10. [10]
    S. M. Nikolsky, Inequalities for entire functions of finite degree and their applications in the theory of differentiable functions of many variables, Proc. Steklov Math. Inst. 38 (1951), 244–278.Google Scholar
  11. [11]
    L. Olsen, Fractal and multifractal dimensions of prevalent measures, Indiana Univ. Math. J. 59 (2010), 661–690.CrossRefzbMATHMathSciNetGoogle Scholar
  12. [12]
    A. Zygmund, Trigonometric Series, 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959.zbMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2014

Authors and Affiliations

  1. 1.Clermont UniversitéUniversité Blaise PascalClerment-FerrandFrance
  2. 2.Laboratoire de MathématiquesCNRS, UMR 6620Aubière CedexFrance

Personalised recommendations