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Studia Mathematica

2009 | 193 | 2 | 109-129

General Dirichlet series, arithmetic convolution equations and Laplace transforms

EN

Abstrakty

EN
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?

109-129

wydano
2009

Twórcy

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