The Analytic Functionals in the Lower Half Plane as a Gel'fand-Shilov Space

Abstract

In this paper we introduce the space (related to the space of GUEL'FAND SHILOV) S+01 of f functions f in (0, + ∞) which satisfy sup tkf(n)(t)| 0 and certain C, A, B > O. We study the expansions of the elements of S+01 and those of its dual (S+01)′ 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+01)′ onto the space of analytic functions in the lower half plane. We deduce that the space S+01 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+βα, are defined in (0, + ∞) in a similar way to the spaces of GEL'FAND-SHILOV (for α, β 0), then, assumed Sβα = {0}, S+βα = {0} holds only for β = 0, α = 1.