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2 results

  • 17 - Probabilistic Tools II

  • from Part 2 - Advanced Toolbox
  • 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 435-472

  • Summary

    We continue the study initiated in Chapter 7 of polynomials with small norms. This time the norm of the polynomial is not taken as the supremum on the n-dimensional polydisc, we take it on B_X, the unit ball of some Banach space. The goal is to show that, given a polynomial, signs can be found in such a way that the norm of the new polynomial, whose coefficients are the original ones multiplied by the signs, has small norm. We do this with three different approaches. The first two approaches use Rademacher random variables as the main probabilistic tools. The first one is based on finding out how many balls of a fixed radius are needed to cover B_X while the second one uses entropy integrals and a good estimate for the entropy numbers of the inclusions between l_p spaces. The third approach is different, and relies on Gaussian random variables, Slepian’s lemma and the fact that Rademacher averages are dominated by Gaussian averages. This approach also allows to get estimates for vector-valued polynomials.

Dirichlet Series and Holomorphic Functions in High Dimensions
  • Andreas Defant, Domingo García, Manuel Maestre, Pablo Sevilla-Peris
  • Published online:
    19 July 2019
    Print publication:
    08 August 2019
  • Over 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.