Convergence and almost sure properties in Hardy spaces of Dirichlet series

Bayart, Frédéric

doi:10.1007/s00208-021-02239-x

Cited by 7 publications

(2 citation statements)

References 19 publications

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“…The following result shows that the key magnitude is in fact k−1 2k L(λ) and extends this to other values of p and q, also answering [2,Question 5.7].…”

supporting

confidence: 71%

Vector-valued general Dirichlet series

et al. 2021

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A. For a general Dirichlet series ∑ a n e −λ n s with frequency λ = (λ n ) n , we study how horizontal translation (i.e. convolution with a Poisson kernel) improves its integrability properties. We characterize hypercontractive frequencies in terms of their additive structure answering some questions posed by Bayart. We also provide sharp bounds for the strips S p (λ) that encode the minimum translation necessary for series in the Hardy space H p (λ) to have absolutely convergent coefficients.

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“…The following result shows that the key magnitude is in fact k−1 2k L(λ) and extends this to other values of p and q, also answering [2,Question 5.7].…”

supporting

confidence: 71%

Vector-valued general Dirichlet series

et al. 2021

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“…They used this revolutionary insight to prove that Bohr's upper estimate indeed is optimal: S = 1/2. In retrospect, one may in the work of Bohr for arbitrary frequency λ = (λ n ) (see, e.g., [9], [48], [49], [50], [51] and [89]). Making the jump from the ordinary case λ = (log n) to arbitrary frequencies reveals serious difficulties.…”

Section: Dirichlet Polynomialsmentioning

confidence: 99%

Projection constants for spaces of multivariate polynomials

Defant¹,

Galicer²,

Mansilla³

et al. 2022

Preprint

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Introduction Chapter 1. General preliminaries Chapter 2. Trigonometric polynomials on abelian groups 2.1. Basics on topological groups 2.2. Rudin's averaging technique 2.3. Integral formula 2.4. The Lozinski-Kharishiladze result revisited Chapter 3. Polynomials on Hilbert spaces 3.1. Orthogonal projection 3.2. Integral formula 3.3. Unitarily invariant index sets 3.4. Estimates Chapter 4. Trace class operators 4.1. Unitary harmonics and their density 4.2. Reproducing kernels 4.3. Minimal projection onto the trace class 4.4. Integral formula Chapter 5. Tensor product methods 5.1. Projection constants and operator ideal norms 5.2. Annihilating coefficients 5.3. Homogeneous building blocks 5.4. Fixing the degree Chapter 6. Unconditionality 6.1. Finite degree vs homogeneous case 6.2. Unconditional basis vs projection constant 6.3. Probabilistic estimates 6.4. Convexity and concavity 6.5. Kadets-Snobar case 6.6. Miscellanea Chapter 7. Dirichlet polynomials 7.1. Integral formula 3 4 CONTENTS 7.2. Applying number theory 7.3. Comparing Sidon constants Chapter 8. Polynomials on the Boolean cube 8.1. Projection constants 8.2. The limit case 8.3. Sidon constants 8.4. Gordon-Lewis cycle Chapter 9. Polynomial projection constants 9.1. Characteristics 9.2. Polynomials on ℓ r 9.3. Multiplication and interpolation 9.4. Characteristics via Lozanovskii's theorem 9.5. Tetrahedral polynomials 9.6. Concrete bounds for characteristics 9.7. Polynomial projection constants vs fundamental functions 9.8. Comparison 9.9. Unconditionalizing the norm Chapter 10. Polynomials on Lorentz and mixed spaces 10.1. The tetrahedral case 10.2. The general case 10.3. An attempt for an approach by interpolation 10.4. Appendix: Mixed spaces Chapter 11. Bohr radii 11.1. Bohr radii vs unconditionality 11.2. Homogenizing sequences 11.3. Bohr radii vs concavity and convexity 11.4. Bohr radii vs projection constants 11.5. Lorentz spaces

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A note on Bohr’s theorem for Beurling integer systems

Broucke,

Kouroupis,

Perfekt

2023

Math. Ann.

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Given a sequence of frequencies $$\{\lambda _n\}_{n\ge 1}$$ { λ n } n ≥ 1 , a corresponding generalized Dirichlet series is of the form $$f(s)=\sum _{n\ge 1}a_ne^{-\lambda _ns}$$ f ( s ) = ∑ n ≥ 1 a n e - λ n s . We are interested in multiplicatively generated systems, where each number $$e^{\lambda _n}$$ e λ n arises as a finite product of some given numbers $$\{q_n\}_{n\ge 1}$$ { q n } n ≥ 1 , $$1 < q_n \rightarrow \infty $$ 1 < q n → ∞ , referred to as Beurling primes. In the classical case, where $$\lambda _n = \log n$$ λ n = log n , Bohr’s theorem holds: if f converges somewhere and has an analytic extension which is bounded in a half-plane $$\{\Re s> \theta \}$$ { ℜ s > θ } , then it actually converges uniformly in every half-plane $$\{\Re s> \theta +\varepsilon \}$$ { ℜ s > θ + ε } , $$\varepsilon >0$$ ε > 0 . We prove, under very mild conditions, that given a sequence of Beurling primes, a small perturbation yields another sequence of primes such that the corresponding Beurling integers satisfy Bohr’s condition, and therefore the theorem. Applying our technique in conjunction with a probabilistic method, we find a system of Beurling primes for which both Bohr’s theorem and the Riemann hypothesis are valid. This provides a counterexample to a conjecture of H. Helson concerning outer functions in Hardy spaces of generalized Dirichlet series.

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Convergence and almost sure properties in Hardy spaces of Dirichlet series

Cited by 7 publications

References 19 publications

Vector-valued general Dirichlet series

Vector-valued general Dirichlet series

Projection constants for spaces of multivariate polynomials

A note on Bohr’s theorem for Beurling integer systems

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