Laguerre Expansions of Gel′fand-Shilov Spaces Original Research Article

The subspaces Gα, Gβ, and Gβα (α, β ≥ 0)of Schwartz′ space S+ in (0, + ∞) are associated with the Hankel transform in the same way as the Gel′fand-Shilov spaces Sα, Sβ, and Sβα are associated with the Fourier transform. Indeed, if we consider the Hankel transform Hγ (γ < −1) defined by Hγ(ƒ)(t) = 12∫∞0 (xt)−γ/2xγJγ([formula]) ƒ(x) dx then Hγ is an isomorphism from Gα, Gβ, and Gβα onto Gα, Gβ, and Gαβ respectively. So. the spaces Gαα are invariant for Hγ. In this paper, we characterize the spaces Gαα (α > 1) in terms of their Fourier-Laguerre coefficients. Also, we characterize the range of the Fourier-Laplace operator FD defined by FD(ƒ)(w) = ∫∞0 ƒ(t) e−(1/2)((1 + w)/(1 − w))t for w ∈ D = {w ∈ C : |w| ≤ 1} when it acts on the space Gαα.