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 / = (0, oo) and characterized by their asymptotic behaviour near the boundaries, we show that: (I) The pseudo-differential operator (-x~lD)" , v e R, D = d/dx , is an automorphism (in the topological sense) on F ; (II) (-x~lD)u is almost an inverse of the Hankel transform hv in the sense that hl/o(x-xD)v(<p) = hfj((p), VpeF, V¡/El; (III) (—x~lD)r has a Fourier-Bessel series representation on a subspace Fb C F and also on its dual F¿ .