General Dirichlet series, arithmetic convolution equations and Laplace transforms

Helge Glöckner ¹   ; Lutz G. Lucht ²   ; Štefan Porubský ³

¹ Institut für Mathematik Universität Paderborn Warburger Str. 100 33098 Paderborn, Germany
² Siemensstr. 1 38640 Goslar, Germany
³ Institute of Computer Science Academy of Sciences of the Czech Republic Pod Vodárenskou věží 2 182 07 Praha 8, Czech Republi

Studia Mathematica, Tome 193 (2009) no. 2, pp. 109-129

Résumé

In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions $g\colon\mathbb N\to\mathbb C$ to convolution equations of the form \[ a_d*g^{*d}+a_{d-1}*g^{*(d-1)}+\cdots+a_1*g+a_0=0, \] where $a_0,\ldots,a_d\colon\mathbb N\to\mathbb C$ 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 $\sum_{x\in X}f(x)e^{-sx}$ ($s\in \mathbb C^k$), where $X\subseteq [0,\infty)^k$ is an additive subsemigroup. If $X$ is discrete and a certain solvability criterion is satisfied, we determine solutions by an elementary recursive approach, adapting an idea of Fečkan [Proc. Amer. Math. Soc. 136 (2008)]. 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 $C\subseteq \mathbb R^k$. 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?

Détail

DOI : 10.4064/sm193-2-2

Keywords: earlier paper proc amer math soc studied solutions colon mathbb mathbb convolution equations form d*g *d d *g * d cdots *g where ldots colon mathbb mathbb given arithmetic functions associated dirichlet series which converge right half plane required function article extend previous results multidimensional general dirichlet series form sum e sx mathbb where subseteq infty additive subsemigroup discrete certain solvability criterion satisfied determine solutions elementary recursive approach adapting idea kan proc amer math soc solution general leads comprehensive question additive subsemigroup pointed closed convex cone subseteq mathbb complex radon measure whose laplace transform satisfies given polynomial equation whose coefficients laplace transforms measures