Dual of the function algebra 𝐴^{-∞}(𝐷) and representation of functions in Dirichlet series

Abstract

In this paper we present the following results: a description, via the Laplace transformation of analytic functionals, of the dual to the (DFS)-space A − ∞ ( D ) A^{-\infty }(D) ( D D being either a bounded C 2 C^2 -smooth convex domain in C N \mathbb {C}^N , with N > 1 N>1 , or a bounded convex domain in C \mathbb {C} ) as an (FS)-space A D − ∞ A^{-\infty }_D of entire functions satisfying a certain growth condition; an explicit construction of a countable sufficient set for A D − ∞ A^{-\infty }_D ; and a possibility of representating functions from A − ∞ ( D ) A^{-\infty }(D) in the form of Dirichlet series.