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The Hölder exponent of some Fourier series

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Abstract

In this paper we study the local regularity of fractional integrals of Fourier series using several definitions of the Hölder exponent. We especially consider series coming from fractional integrals of modular forms. Our results show that in general, cusp forms give rise to pure fractals (as opposed to multifractals). We include explicit examples and computer plots.

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Acknowledgments

We are deeply indebted to Carlos Pastor for his interesting comments on a previous version of this work. In particular, he pointed us the problems with [9, Proposition 1] in the integral case and is currently working in some generalizations that would extend the ranges in the present paper. We also want to thank the anonymous referees for their careful proofreading and suggestions. Fernando Chamizo and Serafín Ruiz-Cabello were partially supported by the Grant MTM2011-22851 of the Ministerio de Ciencia e Innovación (Spain).

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Authors and Affiliations

  1. Department of Mathematics, Universidad Autónoma de Madrid and ICMAT, 28049, Madrid, Spain

    Fernando Chamizo

  2. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111, Bonn, Germany

    Izabela Petrykiewicz

  3. Department of Mathematics, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Serafín Ruiz-Cabello

Authors
  1. Fernando Chamizo
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  2. Izabela Petrykiewicz
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  3. Serafín Ruiz-Cabello
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Corresponding author

Correspondence to Fernando Chamizo.

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Communicated by Stéphane Jaffard.

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Chamizo, F., Petrykiewicz, I. & Ruiz-Cabello, S. The Hölder exponent of some Fourier series. J Fourier Anal Appl 23, 758–777 (2017). https://doi.org/10.1007/s00041-016-9488-4

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  • DOI: https://doi.org/10.1007/s00041-016-9488-4

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