Dirichlet series and convolution equations
type: Item
instance of: scholarly article
title: Dirichlet series and convolution equations
zbMATH Open document ID: 0668.30005
author: D. C. Struppa
author: Carlos A. Berenstein
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1988-01-01T00:00:00Z
review text: There are two main methods to study analytic continuation properties of Dirichlet series \[ f(s)=\sum_{n\geq 1}a_ ne^{\lambda_ ns}\quad with\quad finite\quad density, \] i.e., \(n/\lambda_ n=0(1)\). One is due to Hadamard and consists of the use of entire functions \(\gamma\) that interpolate the values \(a_ n\) at the points \(z=\lambda_ n\) (cf. \textit{V. Bernstein}, Dirichlet series, Gauthier-Villars (1932)). The other is due to S. Mandelbrojt (Hermann (1936), Gauthier-Villars (1952)), it can be interpreted as that of considering the sequence \(\{a_ n\}\) as an element of the dual of a quotient space of a space of holomorphic functions. This is just another way of saying that f satisfies a convolution equation. We use this latter point of view to obtain theorems about Dirichlet series of \(\infty\) density, like \(\lambda_ n\sim \sqrt{n}\). Further work in this direction is in progress by the authors, relating singularity properties of Dirichlet series to vanishing theorems of cohomology. The case of \(\lambda_ n=\log n\) appear in the Ph.D thesis of R. Heymann. Our work is also related to recent work of Dzhrbashyan and of Arakelyan.
Mathematics Subject Classification ID: 30B50
Mathematics Subject Classification ID: 42A75
Mathematics Subject Classification ID: 32A45
zbMATH DE Number: 4093825
zbMATH Keywords: analytic continuation
zbMATH Keywords: mean periodic functions
zbMATH Keywords: Dirichlet series
reviewed by: Carlos A. Berenstein
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195174696
OpenAlex ID: W2043069701
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DOI: 10.2977/PRIMS/1195174696
description: scientific article; zbMATH DE number 4093825