The Stieltjes transform

Abstract

Introduction.The Stieltjes transform is defined by the equation da (t) .fR da(t) C da(t) rHd<We assume that a(t) is of bounded variation in (0, R) for every positive R and that the limit (1) exists.If a(t) is the integral of a function <p(t), we obtain the special case(2) /(*)= -^-dt. Jo x + tIn this paper we discuss two distinct but related questions, the inversion problem and the solubility problem.In the former we assume that f(x) is a function which admits the representation (1) or ( 2), and seek to determine a(t) or <j>(t) from f(x).In the latter we seek necessary and sufficient conditions onf(x) that it should have the representation (1) or (2).A solution of the inversion problem was given by Stieltjesf himself by means of contour integration.His result was a(t +) + a(t -) 1 f-iT ---= lim R -I f(s)ds,