On the Stieltjes transform of generalized functions

Abstract

If f(t) belongs to L(0, R) for every positive R and is such that the integralconverges for x > 0, then F(s) exists for complex s(s ╪ 0) not lying on the negative real axis andfor any positive ξ at which f(ξ+) and f(ξ−) both exist.We define an operator Lk, t[F(x)]byUnder the above conditions on f(t), it is known that for all points t of the Lebesgue set for the function f(t),Let Ln, x denote the differentiation operatorSuppose thatconverges for some x¬ 0; then, if f(t) belongs to L(R−1, R) for every R>1,

An inversion theorem for the Stieltjes transform of distributions

“…If we put ), x>0 ieN 0 , from the proof of Lemma 2 it follows that we have to put In this case we obtain formulae similar to (9)- (11). If (S p f)+eL{ x , then (S p /) + is a regular distribution.…”

“…If we compare our Inversion Theorem with the corresponding ones in Pandey [11] and Pathak [12], for example, we notice the following: the set S of restrictions of functions in £f = Sf(U) on (0, oo) with the topology induced by that of y is a proper subspace of the testing function space S a (0, oo) in their notation. Hence the restrictions of elements from S'JO, oo) belong to S'.…”

“…But in these spaces the differentiation is not defined. This implies that, for example, the Stieltjes transform of d lk) (x -a), a^O, keN, is meaningless in the sense of [16], [10], [11], [5]. (The space investigated in [1] does not contain 8 ik) {x -a), a^O, as well.)…”

Recent developments on the Stieltjes transform of generalized functions

“…THE STIELTJES TRANSFORM OF GENERALIZED FUNCTIONS. Pandey (1972) introduced the Stieltjes transform F(z) of generalized function f(t)C S'(1) by -I F(z) < f(t), (z+t) > (3.3) for z lying in the complex plane with a cut along the negative real axis.…”

Complex inversion formula for the distributional Stieltjes transform

“…There are several approaches to the Stieltjes transform of generalized functions ( [1,10,5,6,3,2]). In this paper we use the definition of the distributional Stieltjes transform of index p (peU\ ( -N o ); M o = Nu{0}), S p -transform, given by Lavoine and Misra [3].…”