The Stieltjes Transform

The Iterated Stieltjes Transform.

THEOREM 4. If Q is a Lie algebra of type All then n > 2 and $ SJ' where J is an i.a.a. of second kind in a simple algebra [. Thus a complete classification of Lie algebras of type A,, depends on the classification of simple associative algebras of second kind and of i.a.a.'s relative to cogredience in these algebras. It can be shown that the latter problem is essentially one of ordinary cogredi...

Extending the Stieltjes Transform

The classical Stieltjes transform is extended to a subspace of Boehmians. The transform is shown to be an analytic function in the half-plane Re z > 0. Some Abelian type theorems are established.

The Generalized Stieltjes Transform and Its Inverse

The generalized Stieltjes transform (GST) is an integral transform that depends on a parameter ρ > 0. In previous work a convenient form of the inverse transformation was derived for the case ρ = 3/2. This paper generalizes that result to all ρ > 0. It is a well-known fact that the GST can be formulated as an iterated Laplace transform, and that therefore its inverse can be expressed as an iter...

Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform

The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) → −1 as y → ∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics.

Quasiasymptotic Expansion of Distributions with Applications to the Stieltjes Transform