Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables

Defant, Andreas; Maestre, Manuel; Prengel, Christopher

doi:10.1515/crelle.2009.068

Cited by 28 publications

(51 citation statements)

References 44 publications

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“…The proof of the last inequality follows exactly the same steps as in Section 3, using (12). To prove the first equality, let us choose r > T 1 (v) and show that r ≥ 1/BH 1 (v) .…”

Section: A Defant and P Sevilla-perismentioning

confidence: 93%

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Convergence of Dirichlet polynomials in Banach spaces

Defant

Sevilla‐Peris²

2011

Trans. Amer. Math. Soc.

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Abstract. Recent results on Dirichlet seriesn a n 1 n s , s ∈ C, with coefficients a n in an infinite dimensional Banach space X show that the maximal width of uniform but not absolute convergence coincides for Dirichlet series and for m-homogeneous Dirichlet polynomials. But a classical non-trivial fact due to Bohnenblust and Hille shows that if X is one dimensional, this maximal width heavily depends on the degree m of the Dirichlet polynomials. We carefully analyze this phenomenon, in particular in the setting of p -spaces.

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“…The proof of the last inequality follows exactly the same steps as in Section 3, using (12). To prove the first equality, let us choose r > T 1 (v) and show that r ≥ 1/BH 1 (v) .…”

Section: A Defant and P Sevilla-perismentioning

confidence: 93%

“…It is not in general true that (6) converges pointwise to P . A deep and complete study of when this happens in the scalar-valued case can be found in [12].…”

Section: Preliminariesmentioning

confidence: 99%

Convergence of Dirichlet polynomials in Banach spaces

Defant

Sevilla‐Peris²

2011

Trans. Amer. Math. Soc.

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“…4-it is mainly based on a fundamental result from local Banach space theory due to Maurey and Pisier [15] (see the preliminaries) combined with ideas from [4,5,8].…”

Section: Corollary 1 For All Finite Dimensional Banach Spaces Y We Hamentioning

confidence: 99%

Bohr’s strip for vector valued Dirichlet series

et al. 2008

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Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet series a n /n s , s ∈ C, converges uniformly but not absolutely, is at most 1/2, and Bohnenblust-Hille that this bound in general is optimal. We prove that for a given infinite dimensional Banach space Y the width of Bohr's strip for a Dirichlet series with coefficients a n in Y is bounded by 1 − 1/ Cot(Y ), where Cot(Y ) denotes the optimal cotype of Y . This estimate even turns out to be optimal, and hence leads to a new characterization of cotype in terms of vector valued Dirichlet series. Mathematics Subject Classification (2000)Primary 32A05; Secondary 46B07 · 46B09 · 46G20 The first, second and third authors were supported by MEC and FEDER Project MTM2005-08210. A. Defant

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“…Very recently a thorough study on domains of convergence of monomial expansions of holomorphic functions has been made in [13]; it contains the following result that will be a key ingredient to us.…”

Section: Introductionmentioning

confidence: 99%