Convergence of Dirichlet polynomials in Banach spaces

Defant, Andreas; Sevilla-Peris, Pablo

doi:10.1090/S0002-9947-2010-05146-3

Convergence of Dirichlet polynomials in Banach spaces
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by Andreas Defant and Pablo Sevilla-Peris PDF

Trans. Amer. Math. Soc. 363 (2011), 681-697 Request permission

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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