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The Analytic Functionals in the Lower Half Plane as a Gel'fand‐Shilov Space
Author(s) -
Duran Antonio J.
Publication year - 1991
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19911530115
Subject(s) - laguerre polynomials , mathematics , orthonormal basis , space (punctuation) , fourier transform , isomorphism (crystallography) , plane (geometry) , analytic function , mathematical analysis , hardy space , pure mathematics , combinatorics , physics , geometry , crystallography , quantum mechanics , linguistics , philosophy , chemistry , crystal structure
In this paper we introduce the space (related to the space of GUEL'FAND SHILOV) S +0 1 of f functions f in (0, + ∞) which satisfy sup t k f ( n ) ( t )| < CA k B n k k for all n, k > 0 and certain C, A, B > O. We study the expansions of the elements of S +0 1 and those of its dual ( S +0 1 )′ with respect to the LAGUERRE orthonormal system, characterizing the sequences of FOURIER‐LAGUERRE coefficients which appear in these expansions. Also, we study the FOURIER transform, proving that it is an isomorphism from the space ( S +0 1 )′ onto the space of analytic functions in the lower half plane. We deduce that the space S +0 1 can be obtained by applying the FOURIER transform to the analytic functionals in the lower half plane. Some applications are given. Finally, we prove that if the spaces S +β α , are defined in (0, + ∞) in a similar way to the spaces of GEL'FAND‐SHILOV (for α, β 0), then, assumed S β α = {0}, S +β α = {0} holds only for β = 0, α = 1.