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1 results

  • 11 - Hardy Spaces of Dirichlet Series

  • from Part 1 - Bohr’s Problem and Complex Analysis on Polydiscs
  • Andreas Defant, Carl V. Ossietzky Universität Oldenburg, Germany, Domingo García, Universitat de València, Spain, Manuel Maestre, Universitat de València, Spain, Pablo Sevilla-Peris
  • Book:
    Dirichlet Series and Holomorphic Functions in High Dimensions
    Published online:
    19 July 2019
    Print publication:
    08 August 2019, pp 268-288

  • Summary

    For each 1 ≤ p ≤ ∞, the Hardy space \mathcal{H}_p of Dirichlet series is defined as the image through the Bohr transform of the Hardy space of functions on the infinite-dimensional polytorus. The Dirichlet polynomials are dense in \mathcal{H}_p for every 1 ≤ p < ∞. For p=2 this coincides with the space of Dirichlet series whose coefficients are square-summable. A Dirichlet series with coefficients a_n belongs to\mathcal{H}_p if and only if the series with coefficients a_n/n^ε is in \mathcal{H}_p for every ε >0 and the norms are uniformly bounded. In this case, the series is the limit as ε tends to 0. As a technical tool to see this, vector-valued Dirichlet series (that is, series with coefficients in some Banach space) are introduced, and some basic definitions and properties (such as the convergence abscissas, Bohr-Cahen formulas) are given.