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
8 - Multidimensional Bohr Radii
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
A holomorphic function f on the disc has a Taylor expansion with coefficients c_k. Bohr asked about the maximal 0<r<1 so that the supremum for |z|<r of ∑ | c_k z^k | is less than or equal to the supremum for |z|<1 of |f(z)|. Bohr’s power series theorem answers this question showing that r=1/3 is best possible. The n-th Bohr radius K_n is defined as the best r for which an analogous question holds for holomorphic functions on the n-dimensional polydisc. The sequence (K_n) is decreasing and tends to 0 as n goes to ∞ asymptotically like (\log n/n)^(1/2). The proof os this relies on an improved version of the polynomial Bohnenblust-Hille inequality (see Chapter 6), where the constant grows at most exponentially, and to get this a Khinchin-Steinhaus inequality for polynomials is needed, showing that all L_p norms of polynomials in n variables are equivalent.
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- Dirichlet Series and Holomorphic Functions in High Dimensions , pp. 181 - 204Publisher: Cambridge University PressPrint publication year: 2019