Stieltjes-generated moment functionals and orthogonality: a transfer principle and an image theorem

Abstract

We study the orthogonality induced by Stieltjes derivatives from the point of view of ordinary polynomial moment functionals. If \(g\) is a derivator and the Stieltjes monomials \(g_{x_0,n}\) are linearly independent, the map \(Φ(t^n)=g_{x_0,n}\) is an isomorphism of differential algebras onto the ordinary polynomial algebra, and the intrinsic \(\star\)-bilinear form on \(g\)-polynomials is the pull-back of a single ordinary moment functional \(\mathcal L\). The object of interest is therefore not a new polynomial algebra but the functional generated by the derivator, and our main theorem determines exactly which functionals occur. Let \(c_-\) and \(c_+\) be the increments of the continuous part of \(g\) on the two sides of the centre, and let \(A\) and \(B\) be the multisets of jump sizes to its right and to its left. Then \(\mathcal L\) is Lebesgue measure on \([-c_-,c_+]\), plus an endpoint derivative functional governed by the elementary symmetric functions of \(A\), plus an absolutely continuous left tail whose density is the survival function of a sum of independent exponentials with means in \(B\); the positions of the jumps disappear. Every quadruple \((c_-,c_+,A,B)\) occurs and is recovered from \(\mathcal L\), so the fibres are described exactly, and \(\mathcal L\) is representable by a positive measure precisely when \(|A|\le1\). A weighted lattice extension, in which the transported moments are the factorial moments of the weight, is treated separately.