Article
Ufa Mathematical Journal
Volume 12, Number 4, pp. 78-89
On Fourier-Laplace transform of a class of generalized functions
Musin I.Kh.
DOI:10.13108/2020-12-4-78
Download PDF
Article on MathNetAbstact
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.