Skip to main content
SpringerLink
Log in

On universality and convergence of the Fourier series of functions in the disc algebra

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We construct functions in the disc algebra whose Fourier series are pointwise universal on countable and dense sets and their sets of divergence contain Gδ and dense sets and have Hausdorff dimension zero. We also see that some classes of closed sets of measure zero do not accept uniformly universal Fourier series, although all such sets accept divergent Fourier series.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. F. Bayart and Y. Heurteaux, Multifractal analysis for the divergence of Fourier series: The extreme cases, J. Anal. Math. 124 (2014), 387–408.

    Article  MathSciNet  MATH  Google Scholar 

  2. G. Herzog and P. C. Kunstmann, Universally divergent Fourier series via Landau’s extremal functions, Comment. Math. Univ. Carolinae 56 (2015), 159–168.

    MathSciNet  MATH  Google Scholar 

  3. Y. Katznelson, Introduction to Harmonic Analysis, 3rd edidtion, Cambridge Univ. Press, Cambridge, 2004.

    Book  MATH  Google Scholar 

  4. J. P. Kahane, Baire’s category theorem and trigonometric series, J. Anal. Math. 80 (2000), 143–182.

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Katsoprinakis, V. Nestoridis, and C. Papachristodoulos, Universality and Cesaro summability, Comp. Methods and Function Theory 12 (2012), 419–448.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. W. Körner, Kahane’s Helson curve, J. Fourier Anal. Appl. 1 (1995), 325–346.

    MathSciNet  MATH  Google Scholar 

  7. J. Müller, Continuous functions with universally divergent Fourier series on small subsets of the circle, C. R. Acad. Sci. Paris, Ser. I 348 (2010), 1155–1158.

    Article  MathSciNet  MATH  Google Scholar 

  8. V. Nestoridis, Universal Taylor series, Ann. Inst. Fourier 46, (1996), 1293–1306.

    Google Scholar 

  9. J. Oxtoby, Measure and Category, 2nd edition, Springer-Verlag Inc., New York, 1980.

    Book  MATH  Google Scholar 

  10. A. Zygmund, Trigonometric Series, Vol. I, 3rd edition, Cambridge Univ. Press, Cambridge, 2002.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Crete, 70013, Iraklio, Crete, Greece

    C. Papachristodoulos & M. Papadimitrakis

Authors
  1. C. Papachristodoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Papadimitrakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Papadimitrakis.

Additional information

Supported by the project 3083-FOURIERDIG which is implemented under the “Aristeia II” Action of the “Operational Programme Education and Lifelong Learning” and is co-founded by the European Social Fund (ESF) and National Resources.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Papachristodoulos, C., Papadimitrakis, M. On universality and convergence of the Fourier series of functions in the disc algebra. JAMA 137, 57–71 (2019). https://doi.org/10.1007/s11854-018-0065-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-018-0065-4

Search

Navigation