Complex inversion for the generalized convolution transformation

An extension of Haimo’s form of Hankel convolutions

“…The fact that (31) is true follows at once in view of (21), and that the expressions in (31) are each equal to (32) is an immediate consequence of (21) In view of asymptotic orders of G m (x, y) 9 (m -0,1, 2, 3, •) for fixed x and large y as stated in (15) and using the technique of the Riemann sum, (36) is proved in a way exactly similar to that employed in [4,Th. 2].…”

“…The Hirschman-Widder convolution transformation [3] has recently been extended to certain classes of generalized functions [7], [4] and their inversion formulae [3;p. 231] have been shown to be still valid when the limiting operation in those formulae is understood as weak convergence in the space D' of Schwartz distributions [5].…”

“…The notation and terminology of this work follows that of [7], [4]. /denotes the open interval (0, oo) and all testing functions herein are defined on I.…”

Complex inversion theorems for generalized Stieltjes transforms

“…_ 0 47T J_ K They also gave a formula, more complicated, for the case -1 < p < 0. Pandey and Zemanian [4] extended Hirschman and Widder's general complex inversion theory to generalized functions.…”

“…[4] Generalized Stieltjes transforms 331 which relates the generalized with the ordinary Stieltjes transform. Inverting (2.2) formally by Theorem A, and using (2.1),…”