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Journal d'Analyse Mathématique

, Volume 137, Issue 1, pp 429–447 | Cite as

Fatou and brothers Riesz theorems in the infinite-dimensional polydisc

  • Alexandru Aleman
  • Jan-Fredrik OlsenEmail author
  • Eero Saksman
Article
  • 30 Downloads

Abstract

We study the boundary behavior of functions in the Hardy spaces on the infinite-dimensional polydisc. These spaces are intimately related to the Hardy spaces of Dirichlet series. We exhibit several Fatou and Marcinkiewicz- Zygmund type theorems for radial convergence of functions with Fourier spectrum supported on \(\mathbb{N}_0^\infty\cup(-\mathbb{N}_0^\infty)\). As a consequence one obtains easy new proofs of the brothers F. and M. Riesz Theorems in infinite dimensions, as well as being able to extend a result of Rudin concerning which functions are equal to the modulus of an H1 function almost everywhere to \(\mathbb{T}^\infty\). Finally, we provide counterexamples showing that the pointwise Fatou theorem is not true in infinite dimensions without restrictions to the mode of radial convergence even for bounded analytic functions.

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References

  1. [1]
    H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen Σ a n/n s, Nachr. Akad.Wiss. Göttingen Math.-Phys. Kl. (1913), 441–488.zbMATHGoogle Scholar
  2. [2]
    Brian J. Cole and T.W. Gamelin, Representing measures andHardy spaces for the infinite polydisk algebra, Proc. London Math. Soc. 53 (1986), 112–142.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    J. B. Garnett, Bounded Analytic Functions, Pure and Applied Mathematics, Vol. 96, Academic Press, New York, 1981.Google Scholar
  4. [4]
    H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L2(0, 1), Duke Math. J. 86 (1997), 1–37.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    H. Helson, Compact groups and Dirichlet series, Ark.Mat. 8 (1969), 139–143.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99 (1958), 165–202.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    D. Hilbert, Wesen und Ziele einer Analysis der unendlichvielen unabhängigen Variabeln, Rend. Circ. Mat. Palermo 27 (1909), 59–74.CrossRefzbMATHGoogle Scholar
  8. [8]
    O. Kallenberg, Foundations of Modern Probability, Probability and its Applications (New York), Springer-Verlag, New York, 2002.CrossRefzbMATHGoogle Scholar
  9. [9]
    H. Queffélec and M. Queffélec, Diophantine Approximation and Dirichlet Series, Harish-Chandra Research Institute Lecture Notes, Vol. 2, Hindustan Book Agency, New Delhi, 2013.Google Scholar
  10. [10]
    W. Rudin, Fourier Analysis on Groups, Interscience Tracts in Pure and Applied Mathematics, Vol. 12, Interscience Publishers, New York–London, 1962.Google Scholar
  11. [11]
    W. Rudin, Function Theory in Polydiscs, W. A. Benjamin, New York–Amsterdam, 1969.zbMATHGoogle Scholar
  12. [12]
    E. Saksman and K. Seip, Integral means and boundary limits of Dirichlet series, Bull. Lond. Math. Soc. 41 (2009), 411–422.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    A. Zygmund, Trigonometric Series: Vols. I, II, Cambridge University Press, London, 1968.Google Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  • Alexandru Aleman
    • 1
  • Jan-Fredrik Olsen
    • 1
    Email author
  • Eero Saksman
    • 2
  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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