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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Convergence of Dirichlet polynomials in Banach spaces


Authors: Andreas Defant and Pablo Sevilla-Peris
Journal: Trans. Amer. Math. Soc. 363 (2011), 681-697
MSC (2000): Primary 46B07; Secondary 32A05, 46B09, 46G20, 30B50
Published electronically: September 15, 2010
MathSciNet review: 2728583
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Abstract: Recent results on Dirichlet series $ \sum_{n} a_{n} \frac{1}{n^{s}}$, $ s \in \mathbb{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 $ \ell_p$-spaces.


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Additional Information

Andreas Defant
Affiliation: Institute of Mathematics, Carl von Ossietzky University, D–26111 Oldenburg, Germany
Email: defant@mathematik.uni-oldenburg.de

Pablo Sevilla-Peris
Affiliation: Institute of Mathematics, Carl von Ossietzky University, D–26111 Oldenburg, Germany – and – Departamento de Matemática Aplicada and IUMPA, ETSMRE, Universidad Politécnica de Valencia, Av. Blasco Ibáñez, 21, E–46010 Valencia, Spain
Email: psevilla@mat.upv.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05146-3
PII: S 0002-9947(2010)05146-3
Keywords: Vector valued Dirichlet series, Dirichlet polynomials, Banach spaces
Received by editor(s): July 14, 2008
Received by editor(s) in revised form: March 7, 2009
Published electronically: September 15, 2010
Additional Notes: Both authors were supported by the MEC Project MTM2008-03211. The second author was partially supported by grants PR2007-0384 (MEC) and UPV-PAID-00-07.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.