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2 results

  • 15 - Infinite Dimensional Holomorphy

  • 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 351-409

  • Summary

    We give an introduction to vector-valued holomorphic functions in Banach spaces, defined through Fréchet differentiability. Every function defined on a Reinhardt domain of a finite-dimensional Banach space is analytic, i.e. can be represented by a monomial series expansion, where the family of coefficients is given through a Cauchy integral formula. Every separate holomorphic (holomorphic on each variable) function is holomorphic. This is Hartogs’ theorem, which is proved using Leja’s polynomial lemma. For infinite-dimensional spaces, homogeneous polynomials are defined as the diagonal of multilinear mappings. A function is holomorphic if and only if it is Gâteaux holomorphic and continuous, if and only if it has representation as a series of homogeneous polynomials (known as Taylor expansion). A function is weak holomorphic if the composition with every functional is holomorphic. A function is holomorphic if and only if it is weak holomorphic. Analytic functions are holomorphic.

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.