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

Abstract

Abstract. In this paper we present the following results: a description, via the Laplace transformation of analytic functionals, of the dual to the (DFS)-

Mutual dualities betweenA−∞(Ω) and for lineally convex domains

“…As it will be seen in this article, the duality theorem guarantees an opportunity to represent each function from A 1 ð e Þ in the form of series of partial fractions, while a surprising result, in comparison with [12,13], is that it is impossible to have a similar representation for functions from A À1 (), despite the existence of sufficient sets in its dual space.…”

“…In our recent papers [12,13], for a bounded convex domain of C n , we established, via the Laplace transformation, the mutual dualities between A À1 () and the space A À1 of entire functions in C n with a certain growth condition. We also gave an explicit construction of a countable sufficient set for the dual space and then, applying the so-called 'dual relationship', we showed that any function from either A À1 () or A À1 can always be represented in the form of a Dirichlet series.…”

“…Complex Variables and Elliptic Equations 3 1617 A 1 0 ðC n Þ, via the Cauchy transformation, is valid for strictly starlike Jordan domain in C having only a rectifiable boundary (see in this connection our papers [12,13]). Moreover, we believe that for n ¼ 1 the assumption of starlikeness for is superfluous.…”

Surjectivity criteria for convolution operators inA−∞

“…The proof of Theorem 1.2 is based on [1, Theorem 2.1] (see also [2,3] for more details), giving the description of the strong dual for A −∞ (Ω) via Fourier-Borel transformation as the (FS)-space…”

Minimal absolutely representing systems of exponentials for A−∞(Ω)

“…In [5,Theorem 4.3] it was shown that in the space A −∞ (Ω ) there always exists an absolutely representing system (for short, ARS) of exponentials E Λ = (e λ k z ) ∞ k=1 , where the sequence of complex frequencies Λ := (λ k ) ∞ k=1 ⊂ C satisfies the condition λ k → ∞ as k → ∞. Equivalently, any function f ∈ A −∞ (Ω ) can be represented in the form of a Dirichlet series…”