LAGUERRE EXPANSIONS OF TEMPERED DISTRIBUTIONS AND GENERALIZED-FUNCTIONS

Let (Lnα(t))n be for α > -1, the sequence of generalized Laguerre polynomials. Then we consider the orthonormal system (( n! Γ(n + α + 1)) 1 2 Lnα(t) t α 2 e -t 2)n in L2([0, + ∞)). We study the expansions of certain spaces of generalized functions with respect to this orthonormal system (for α = 0, this space of generalized functions is the space of tempered distributions with positive support). We characterize the sequences of Fourier-Laguerre coefficients which appear in these expansions. Finally we give some applications. © 1990.