On Fourier–Laplace transform of a class of generalized functions

Abstract

We consider a subspace of Schwartz space of fast decaying infinitely differentiable functions on an unbounded closed convex set in a multidimensional real space with a topology defined by a countable family of norms constructed by means of a family ${\mathfrak M}$ of a logarithmically convex sequences of positive numbers. Owing to the mentioned conditions for these sequence, the considered space is a Fréchet–Schwartz one. We study the problem on describing the strong dual space for this space in terms of the Fourier–Laplace transforms of functionals. Particular cases of this problem were considered by by J.W. De Roever in studying problems of mathematical physics, complex analysis in the framework of a developed by him theory of ultradistributions with supports in an unbounded closed convex set; similar studies were also made by by P.V. Fedotova and by the author of the present paper. Our main result, presented in Theorem 1, states that the Fourier–Laplace transforms of the functionals establishes an isomorphism between the strong dual space of the considered space and some space of holomorphic functions in a tubular domain of the form ${\mathbb{R}}^n + iC$, where $C$ is an open convex acute cone in ${\mathbb{R}}^n$ with the vertex at the origin; the mentioned holomorphic functions possess a prescribed growth majorants at infinity and at the boundary of the tubular domain. The work is close to the researches by V. S. Vladimirov devoted to the theory of the Fourier–Laplace transformatation of tempered distributions and spaces of holomorphic functions in tubular domains. In the proof of Theorem 1 we apply the scheme proposed by M. Neymark and B. A. Taylor as well as some results by P. V. Yakovleva (Fedotova) and the author devoted to Paley–Wiener type theorems for ultradistributions.