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Full 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.
Contents
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.
