General Dirichlet series, arithmetic convolution equations and Laplace transforms

Helge Glöckner, Lutz G. Lucht, and Štefan Porubský. "General Dirichlet series, arithmetic convolution equations and Laplace transforms." Studia Mathematica 193.2 (2009): 109-129. <http://eudml.org/doc/286130>.

@article{HelgeGlöckner2009,
abstract = {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?},
author = {Helge Glöckner, Lutz G. Lucht, Štefan Porubský},
journal = {Studia Mathematica},
keywords = {arithmetic function; Dirichlet convolution; polynomial equation; analytic equation; topological algebra; holomorphic functional calculus; implicit function theorem; Laplace transform; semigroup; complex measure},
language = {eng},
number = {2},
pages = {109-129},
title = {General Dirichlet series, arithmetic convolution equations and Laplace transforms},
url = {http://eudml.org/doc/286130},
volume = {193},
year = {2009},
}

TY - JOUR
AU - Helge Glöckner
AU - Lutz G. Lucht
AU - Štefan Porubský
TI - General Dirichlet series, arithmetic convolution equations and Laplace transforms
JO - Studia Mathematica
PY - 2009
VL - 193
IS - 2
SP - 109
EP - 129
AB - 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?
LA - eng
KW - arithmetic function; Dirichlet convolution; polynomial equation; analytic equation; topological algebra; holomorphic functional calculus; implicit function theorem; Laplace transform; semigroup; complex measure
UR - http://eudml.org/doc/286130
ER -