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.

Andreas Defant is Professor of Mathematics at Carl V. Ossietzky Universitat Oldenburg, Germany. Domingo Garcia is Professor of Mathematics at Universitat de Valencia, Spain. Manuel Maestre is Full Professor of Mathematics at Universitat de Valencia, Spain. Pablo Sevilla-Peris is Associate Professor of Mathematics at Universitat Politecnica de Valencia, Spain.

'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