General Dirichlet series, arithmetic convolution equations and Laplace transforms

Helge Glöckner ; Lutz G. Lucht ; Štefan Porubský

Studia Mathematica, Tome 192 (2009), p. 109-129 / Harvested from The Polish Digital Mathematics Library

Résumé

In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form $a_{d}∗g^{∗d} + a_{d-1}∗g^{∗(d-1)} + ⋯ + a₁∗g + a₀ = 0$, where $a₀,...,a_{d}: ℕ → ℂ$ 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 $∑_{x∈X} f(x)e^{-sx}$ ($s ∈ ℂ^{k}$), where $X ⊆ [0,∞)^{k}$ 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 $C ⊆ ℝ^{k}$. 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

Publié le : 2009-01-01

Zbl 1177.11011

EUDML-ID : urn:eudml:doc:286130