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

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 $\mathrm{\Phi }(t^n)=g_{x_0,n}$ 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 $\mathrm{\mid }A\mathrm{\mid }\le 1$. A weighted lattice extension, in which the transported moments are the factorial moments of the weight, is treated separately.