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General Dirichlet Series, Arithmetic Convolution Equations and Laplace Transforms
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SYSNO ASEP 0326688 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title General Dirichlet Series, Arithmetic Convolution Equations and Laplace Transforms Title Obecné Dirichletovy řady, rovnice s aritmetickou konvolucí a Laplaceovy transformace Author(s) Glöckner, H. (DE)
Lucht, L.G. (DE)
Porubský, Štefan (UIVT-O) SAI, RIDSource Title Studia mathematica. - : Polska Akademia Nauk - ISSN 0039-3223
Roč. 193, č. 2 (2009), s. 109-129Number of pages 21 s. Language eng - English Country PL - Poland Keywords arithmetic function ; Dirichlet convolution ; polynomial equation ; analytic equation ; topological algebra ; holomorphic functional calculus ; implicit function theorem ; Laplace transform ; semigroup ; complex measure Subject RIV BA - General Mathematics R&D Projects GA201/07/0191 GA ČR - Czech Science Foundation (CSF) CEZ AV0Z10300504 - UIVT-O (2005-2011) UT WOS 000271387800002 EID SCOPUS 69549092194 DOI 10.4064/sm193-2-2 Annotation In an earlier paper, we studied solutions g to convolution equations of the form a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+...+a_1*g+a_0=0, where a_0, ..., a_d are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form \sum_{x\in X} f(x) e^{-sx} (s in C^k), where X is an additive subsemigroup of [0,\infty)^k. If X is discrete and a certain solvability criterion is satisfied, we determine solutions by an elementary recursive approach, adapting an idea of Feckan. The solution of the general case leads us to a more comprehensive question: Let X be an additive subsemigroup of a pointed, closed convex cone C in R^k. Can we find a complex Radon measure on X whose Laplace transform satisfies a given polynomial equation whose coefficients are Laplace transforms of such measures? Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2010
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