Estimates for vector valued Dirichlet polynomials

Defant, Andreas; Schwarting, Ursula; Sevilla‐Peris, Pablo

doi:10.1007/s00605-013-0600-4

Cited by 38 publications

(11 citation statements)

References 23 publications

(24 reference statements)

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“…We need the following lemma [10, page 492] (see also [12,Lemma 4.3] or [3, Lemma 2.6]). Now we define the following m-homogeneous polynomial in π(x) variables…”

Section: Resultsmentioning

confidence: 99%

The Dirichlet–Bohr radius

et al. 2015

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Abstract. Denote by Ω(n) the number of prime divisors of n ∈ N (counted with multiplicities). For x ∈ N define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial n≤x a n n −s we have n≤x |a n |r Ω(n) ≤ sup t∈R n≤x a n n −it .We prove that the asymptotically correct order of L(x) is (log x) 1/4 x −1/8 . Following Bohr's vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows to translate various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa.

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“…We need the following lemma [10, page 492] (see also [12,Lemma 4.3] or [3, Lemma 2.6]). Now we define the following m-homogeneous polynomial in π(x) variables…”

Section: Resultsmentioning

confidence: 99%

The Dirichlet–Bohr radius

et al. 2015

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“…The following proposition gives some evidence that the estimate in (33) (as in the particular case (32)) might be an equality. Proposition 4.14.…”

Section: 3mentioning

confidence: 94%

“…One of many fruitful lines of research in this respect is given by the analysis of functional analytic aspects of vector-valued ordinary Dirichlet series, so series a n n −s with coefficients a n in a given normed space X. The list [22], [23], [24], [25], [26], [30], [31], [32], [33] of recent articles indeed documents this activity; let us also mention that some of the results proved in these articles are collected in the recent monograph [34,Chapter 26].…”

Section: Introductionmentioning

confidence: 99%

Vector-valued general Dirichlet series

Carando

Defant

Marceca

et al. 2020

Preprint

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Opened up by early contributions due to, among others, H. Bohr, Hardy-Riesz, Bohnenblust-Hille, Neder and Landau the last 20 years show a substantial revival of systematic research on ordinary Dirichlet series a n n −s , and more recently even on general Dirichlet series a n e −λns . This involves the intertwining of classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. Motivated through this line of research the main goal of this article is to start a systematic study of a variety of fundamental aspects of vector-valued general Dirichlet series a n e −λns , so Dirichlet series, where the coefficients are not necessarily in C but in some arbitrary Banach space X.

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“…In the ordinary case λ = (log n) the study of functional analytic aspects of vector-valued Dirichlet series accumulated quite some amount of research, see e.g. [14], [15], [17], [18], [20], [21], [22], or [23]. As an example we recall the following strong extension of (2) from [20]: The maximal width of the strip of uniform, non absolute convergence of X-valued ordinary Dirichlet series is given by the formula…”

Section: Vector-valued Aspects Of General Dirichlet Seriesmentioning

confidence: 99%

Hardy spaces of general Dirichlet series — a survey

Defant

Schoolmann

2019

Banach Center Publ.

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The main purpose of this article is to survey on some key elements of a recent H p -theory of general Dirichlet series a n e −λns , which was mainly inspired by the work of Bayart and Helson on ordinary Dirichlet series a n n −s . In view of an ingenious identification of Bohr, the H p -theory of ordinary Dirichlet series can be seen as a sub-theory of Fourier analysis on the infinite dimensional torus T ∞ . Extending these ideas, the H p -theory of λ-Dirichlet series is build as a sub-theory of Fourier analysis on what we call λ-Dirichlet groups. A number of problems is added.

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Estimates for vector valued Dirichlet polynomials

Abstract: Abstract. We estimate the 1-norm N n=1 an of finite Dirichlet polynomials N n=1 ann −s , s ∈ C with coefficients an in a Banach space. Our estimates quantify several recent results on Bohr's strips of uniform but non absolute convergence of Dirichlet series in Banach spaces.

Cited by 38 publications

References 23 publications

The Dirichlet–Bohr radius

The Dirichlet–Bohr radius

Vector-valued general Dirichlet series

Hardy spaces of general Dirichlet series — a survey

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