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 gx0,n are linearly independent, the map Φ(tn)=gx0,n is an isomorphism of differential algebras onto the ordinary polynomial algebra, and the intrinsic ⋆-bilinear form on g-polynomials is the pull-back of a single ordinary moment functional 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 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 ℒ, so the fibres are described exactly, and ℒ is representable by a positive measure precisely when |A|≤1. A weighted lattice extension, in which the transported moments are the factorial moments of the weight, is treated separately.
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BibTeX
@misc{area2026stieltjesgenerated,
title = {{Stieltjes-generated moment functionals and orthogonality: a transfer principle and an image theorem}},
author = {Iván Area and Francisco J. Fernández and F. Adrián F. Tojo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23031},
eprint = {2610.23031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23031},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stieltjes-generated moment functionals and orthogonality: a transfer principle and an image theorem
AU - Iván Area
AU - Francisco J. Fernández
AU - F. Adrián F. Tojo
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23031
UR - https://arxiv.org/abs/2610.23031
AB - 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 ⋆-bilinear form on g-polynomials is the pull-back of a single ordinary moment functional 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 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 ℒ, so the fibres are described exactly, and ℒ is representable by a positive measure precisely when |A|≤1. A weighted lattice extension, in which the transported moments are the factorial moments of the weight, is treated separately.
ER -