Book contents
- Frontmatter
- Contents
- Introduction
- Part 1 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 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
7 - Probabilistic Tools I
from Part 1 - Bohr’s Problem and Complex Analysis on Polydiscs
Published online by Cambridge University Press: 19 July 2019
- Frontmatter
- Contents
- Introduction
- Part 1 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 2 Advanced Toolbox
- Part 3 Replacing Polydiscs by Other Balls
- Part 4 Vector-Valued Aspects
- References
- Symbol Index
- Subject Index
Summary
We give an alternative, probabilistic, approach to two of the subjects considered so far: the optimality of the exponent in the polynomial Bohnenblust-Hille inequality (see Chapter 6) and the lower bound for S in Bohr’s problem (see Chapters 1 and 4). We use a probabilistic device: the Kahane-Salem-Zygmund inequality. This shows that, for a given finite family of coefficients, a choice of signs can be found in such a way that the polynomial whose coefficients are the original ones multiplied by the signs has small norm (supremum on the polydisc). The proof uses Bernstein’s inequality and Rademacher random variables. We also look at the relationship between Rademacher and Steinhaus random variables, and deduce the classical Khinchin inequality from the Khinchin-Steinhaus inequality (see Chapter 6). We consider Dirichlet series, place signs before the coefficients, and define the almost sure abscissas (in each of the senses from Chapter 1) by considering each convergence for almost every choice of signs. An analogue of Bohr’s problem in this sense is considered.
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- Dirichlet Series and Holomorphic Functions in High Dimensions , pp. 153 - 180Publisher: Cambridge University PressPrint publication year: 2019