Abelian theorems for the distributional Stieltjes transformation

Abstract

0. Introduction. Asymptotic behaviour of distributions (Silva(8) and Benedetto (l)) plays a fundamental role in the analysis of singularities of generalized integral transforms. Such analysis in terms of Abelian theorems for the distributional Stieltjes transformation has been obtained by Misra (6) and Lavoine and Misra (3). To obtain his results Misra used some Abelian theorems (see Misra (6), theorems 3·1·1 and 4·1·1) for the Stieltjes transformation of functions which were essentially generalizations of the results given by Widder (9), pp. 183–185. The object of the present paper is to complete the work given by the authors in (3); Sections 2 and 3 of this paper are devoted to obtaining Abelian theorems concerning the distributional Stieltjes transformation in terms of the behaviour of this transform near the origin and at infinity.