The fourier-bessel series representation of the pseudo-differential operator (−r−1D)v

Abstract

For a certain Fréchet space F consisting of complex-valued C∞functions defined on I= (0, ∞) and characterized by their asymptotic behaviour near the boundaries, we show that: (I) The pseudo-differential operator −r−1D)v v ∈ R, D = d/dx, is an automorphism (in the topological sense) on F; (II) −r−1D)v is almost an inverse of the Hankel transform hv in the sense that(III) −r−1D)v has a Fourier-Bessel series representation on a subspace Fb⊂ F and also on its dual Fb. © 1992 American Mathematical Society.