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2 results

  • 12 - Bohr’s Problem in Hardy Spaces

  • 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 289-314

  • Summary

    Each Hardy space of Dirichlet series \mathcal{H}_p has an associated abscissa, and the analogue to Bohr’s problem arises in a natural way: to determine the maximal distance S_p between this abscissa and the abscissa of absolute convergence. If a Dirichlet series with coefficients (a_n) belongs to \mathcal{H}_p, then the series with coefficients (a_n/n^{ε}) belongs to \mathcal{H}_q for all q>p and ε >0. It is shown that S_p=1/2, and that, if we only consider m-homogeneous Dirichlet series, S_p^m=1/2. For every 1 ≤ p < ∞ the set of monomial convergence of the Hardy space H_p of functions on the infinite dimensional polytorus (hence also of the Hardy space H_2 on the infinite-dimensional polytorus) is l_2 ∩ Bc0. The space of all multipliers on the Hardy space of Dirichlet series \mathcal{H}_p coincides with \mathcal{H}_\infty.

  • PACKING STRIPS IN THE HYPERBOLIC PLANE

  • T. H. Marshall, G. J. Martin
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 46 / Issue 1 / February 2003
    Published online by Cambridge University Press:
    27 January 2003, pp. 67-73
    • Article
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  • A strip of radius $r$ in the hyperbolic plane is the set of points within distance $r$ of a given geodesic. Wedefine the density of a packing of strips of radius $r$ and prove that this density cannot exceed

    $$ \mathcal{S}(r)=\frac{3}{\pi}\sinh r\mathrm{arccosh}\biggl(1+\frac{1}{2\sinh^2r}\biggr). $$

    This bound is sharp for every value of $r$ and provides sharp bounds on collaring theorems for simple geodesics onsurfaces.

    AMS 2000 Mathematics subject classification: Primary 51M09; 52C15. Secondary 51M04