Dirichlet Series and Holomorphic Functions in High Dimensions

Description

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. Cover......Page 1 Front Matter ......Page 2 NEW MATHEMATICAL MONOGRAPHS......Page 3 Dirichlet Series and Holomorphic Functions in High Dimensions......Page 4 Copyright ......Page 5 Contents ......Page 6 Introduction ......Page 12 PART ONE: BOHR’S PROBLEM AND COMPLEX ANALYSIS ON POLYDISCS......Page 30 1 The Absolute Convergence Problem......Page 32 2 Holomorphic Functions on Polydiscs......Page 66 3 Bohr’s Vision......Page 105 4 Solution to the Problem......Page 122 5 The Fourier Analysis Point of View......Page 140 6 Inequalities I......Page 158 7 Probabilistic Tools I......Page 182 8 Multidimensional Bohr Radii......Page 210 9 Strips under the Microscope......Page 234 10 Monomial Convergence of Holomorphic Functions......Page 259 11 Hardy Spaces of Dirichlet Series......Page 297 12 Bohr’s Problem in Hardy Spaces......Page 318 13 Hardy Spaces and Holomorphy......Page 344 PART TWO: ADVANCED TOOLBOX......Page 366 14 Selected Topics on Banach Space Theory......Page 368 15 Infinite Dimensional Holomorphy......Page 380 16 Tensor Products......Page 439 17 Probabilistic Tools II......Page 464 PART THREE: REPLACING POLYDISCS BY OTHER BALLS......Page 502 18 Hardy–Littlewood Inequality......Page 504 19 Bohr Radii in lp Spaces and Unconditionality......Page 515 20 Monomial Convergence in Banach Sequence Spaces......Page 535 21 Dineen’s Problem......Page 560 22 Back to Bohr Radii......Page 584 PART FOUR: VECTOR-VALUED ASPECTS......Page 594 23 Functions of One Variable......Page 596 24 Vector-Valued Hardy Spaces......Page 613 25 Inequalities IV......Page 641 26 Bohr’s Problem for Vector-Valued Dirichlet Series......Page 674 References......Page 693 Symbol Index......Page 706 Subject Index......Page 707