Orthogonality, Spectral Filling, and Uncertainty
Abstract
Let a normalized continuous Parseval frame describe the phase space of a reproducing kernel Hilbert space. The frame coefficients of an orthonormal family have total mass equal to the number of states and pointwise mass at most one. This elementary exclusion principle gives explicit lower bounds for the total cost of a family in terms of the volume of low-cost regions. For radial costs, it yields collective mean-dispersion, umbrella, and transport inequalities. The same argument applies to normalized Bessel families, with the Bessel bound measuring the permitted overlap. We also consider semiclassical families for which the frame measures vary. Weak convergence of the normalized measures, together with averaged localization of the normalized kernels, yields continuous-symbol Szegő limits and sharp asymptotics for the lowest Toeplitz energies. The results apply to Bargmann--Fock and weighted Bergman spaces, Gabor and Paley--Wiener spaces, and orthogonal-polynomial and determinantal ensembles, including the GUE.
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BibTeX
@misc{alvarez2026orthogonality,
title = {{Orthogonality, Spectral Filling, and Uncertainty}},
author = {Travis Alvarez and Trevor Camper and Mishko Mitkovski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02484},
eprint = {2610.02484},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02484},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Orthogonality, Spectral Filling, and Uncertainty
AU - Travis Alvarez
AU - Trevor Camper
AU - Mishko Mitkovski
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02484
UR - https://arxiv.org/abs/2610.02484
AB - Let a normalized continuous Parseval frame describe the phase space of a reproducing kernel Hilbert space. The frame coefficients of an orthonormal family have total mass equal to the number of states and pointwise mass at most one. This elementary exclusion principle gives explicit lower bounds for the total cost of a family in terms of the volume of low-cost regions. For radial costs, it yields collective mean-dispersion, umbrella, and transport inequalities. The same argument applies to normalized Bessel families, with the Bessel bound measuring the permitted overlap. We also consider semiclassical families for which the frame measures vary. Weak convergence of the normalized measures, together with averaged localization of the normalized kernels, yields continuous-symbol Szegő limits and sharp asymptotics for the lowest Toeplitz energies. The results apply to Bargmann--Fock and weighted Bergman spaces, Gabor and Paley--Wiener spaces, and orthogonal-polynomial and determinantal ensembles, including the GUE.
ER -