A Stieltjes representation of order p for $ \mathrm{tr}[(A+tB)^{-p}]$: Heinävaara's measure, B-splines, and Stahl's theorem
Abstract
For A positive definite, B positive semidefinite and p≥1 we show that tr[(A+tB)−p]=∫(1+tℓ)−p dρp(ℓ) for a positive measure ρp on the spectral interval of L=A−1/2BA−1/2, and we identify the measure: it has atoms tr[(PsA−1Ps)p] at the eigenvalues of L and a continuous density, an integral of Heinävaara's tracial joint spectral measure along rays. For integer p the positivity of ρp follows from the positivity of Heinävaara's measure alone, and the density is a sum of Curry-Schoenberg B-splines; for real p the representation is obtained by inverting Heinävaara's transform. For 0<p<1 the function is a Stieltjes function of order p only if [A,B]=0, but a positive representation of order one holds, with an explicit density. Along Ap=pI+A0, p→∞, the measures converge to the measure of Stahl's theorem (the Bessis-Moussa-Villani conjecture) for tr e−(A0+tB).
Related
- A note on the rectangular Schoenberg-Whitney theorem for clamped B-splines
- Neural Wasserstein Splines: Representation-Invariant Second-Order Priors for Trajectory Inference
- Stieltjes-generated moment functionals and orthogonality: a transfer principle and an image theorem
- Generalized Harmonic Measures: Synchronized Approximation, Representation and Quantitative Measure Recovery
- A Liouville theorem for plurisubharmonic functions with periodic complex Monge-Ampère measure
BibTeX
@misc{reybellet2026stieltjes,
title = {{A Stieltjes representation of order $p$ for $ \mathrmtr[(A+tB)^-p]$: Heinävaara's measure, B-splines, and Stahl's theorem}},
author = {Luc Rey-Bellet},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03208},
eprint = {2611.03208},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03208},
}
RIS
TY - UNPB
TI - A Stieltjes representation of order p for $ \mathrm{tr}[(A+tB)^{-p}]$: Heinävaara's measure, B-splines, and Stahl's theorem
AU - Luc Rey-Bellet
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03208
UR - https://arxiv.org/abs/2611.03208
AB - For A positive definite, B positive semidefinite and p≥1 we show that tr[(A+tB)^−p]=∫(1+tℓ)^−p dρ_p(ℓ) for a positive measure ρ_p on the spectral interval of L=A^−1/2BA^−1/2, and we identify the measure: it has atoms tr[(P_sA^−1P_s)^p] at the eigenvalues of L and a continuous density, an integral of Heinävaara's tracial joint spectral measure along rays. For integer p the positivity of ρ_p follows from the positivity of Heinävaara's measure alone, and the density is a sum of Curry-Schoenberg B-splines; for real p the representation is obtained by inverting Heinävaara's transform. For 0<p<1 the function is a Stieltjes function of order p only if [A,B]=0, but a positive representation of order one holds, with an explicit density. Along A_p=pI+A_0, p→∞, the measures converge to the measure of Stahl's theorem (the Bessis-Moussa-Villani conjecture) for tr e^−(A_0+tB).
ER -