Dirichlet Series and Holomorphic Functions in High Dimensions

Using contemporary concepts, this book describes the interaction between Dirichlet series and holomorphic functions in high dimensions.

Andreas Defant (Author), Domingo García (Author), Manuel Maestre (Author), Pablo Sevilla-Peris (Author)

9781108476713, Cambridge University Press

Hardback, published 8 August 2019

706 pages, 3 b/w illus.
23.4 x 15.7 x 4.2 cm, 1.14 kg

'Dirichlet series have been studied for well over a century and still form an integral part of analytic number theory … The purpose of this text is to illustrate the connections between the Dirichlet series per se and the fields just mentioned, e.g., both functional and harmonic analysis … The authors succeed in transferring important concepts and theorems of analytic function theory, in finitely many variables, to the theory in infinitely many variables.' J. T. Zerger, Choice

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.

Introduction
Part I. Bohr's Problem and Complex Analysis on Polydiscs: 1. The absolute convergence problem
2. Holomorphic functions on polydiscs
3. Bohr's vision
4. Solution to the problem
5. The Fourier analysis point of view
6. Inequalities I
7. Probabilistic tools I
8. Multidimensional Bohr radii
9. Strips under the microscope
10. Monomial convergence of holomorphic functions
11. Hardy spaces of Dirichlet series
12. Bohr's problem in Hardy spaces
13. Hardy spaces and holomorphy
Part II. Advanced Toolbox: 14. Selected topics on Banach space theory
15. Infinite dimensional holomorphy
16. Tensor products
17. Probabilistic tools II
Part III. Replacing Polydiscs by Other Balls: 18. Hardy–Littlewood inequality
19. Bohr radii in lp spaces and unconditionality
20. Monomial convergence in Banach sequence spaces
21. Dineen's problem
22. Back to Bohr radii
Part IV. Vector-Valued Aspects: 23. Functions of one variable
24. Vector-valued Hardy spaces
25. Inequalities IV
26. Bohr's problem for vector-valued Dirichlet series
References
List of symbols
Subject index.

Subject Areas: Calculus & mathematical analysis [PBK]