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.

Quasiasymptotic expansion of distributions from S+′ and the asymptotic expansion of the distributional Stieltjes transform

“…Some advantages of this definition were mentioned in [8]. It is proved in [7] (iii) (Srf)(x)-ci F(r+l) I i=l PROOF. We shall prove the theorem by using the similar idea as in the proof of the main theorem in [9].…”

“…In the papers [9,10] authors followed the definition of the distributional Stieltjes transform given in [7] which enabled them to use the strong theory of the space of tempered distributions S'. In fact, they generalized slightly the definition of Lavoine and '= {fe Misra.…”

“…In fact, they generalized slightly the definition of Lavoine and '= {fe Misra. Using the notion of the quasiasymptotic behaviour of distributions from S+ supp f [0,)}, introduced by Zavialov in [15], they obtained more general results than in [6,7,2] for the asymptotic behaviour of the distributional Stieltjes transform at and 0+. Let us notice that the notion of the quasiasymptotic behaviour of distributions was studied by Droinov, Vladimiraov and Zavialov in several papers (see [13] and references there) in which they obtained remarkable results in the quantum flela theory.…”

“…Following [4] we define the open q.a.e. [7]. Some advantages of this definition were mentioned in [8].…”

Tauberian theorem for the distributional Stieltjes transformation

“…The quasiasymptotic behaviour at infinity of tempered distributions with supports in [0, ) (denoted by S$) was defined by Zavijalov (see for instance [I] We use the definition of the distributional Stieltjes transform given in [2], [3] By [6] it is …”