Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.02484

Orthogonality, Spectral Filling, and Uncertainty

Travis Alvarez·Trevor Camper·Mishko Mitkovski
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Travis Alvarez, Trevor Camper, Mishko Mitkovski. 2026-10-01. Orthogonality, Spectral Filling, and Uncertainty. https://arxiv.org/abs/2610.02484

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality

Working at the level of a sectorial operator $A$ on a Banach space $X$ and an unbounded operator $\partial$ defined on a subspace of $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\|A^αx\|_X \lesssim \|\partial x\|_Y$ for some $0 < α< 1$, combined with the condition $0 \in ρ(A_0)$, where $A_0$ is the part of $A$ on the closure of the range of $A$, implies the Poincaré inequality $\|x - P(x)\|_X \lesssim \|\partial x\|_Y$, where $P$ is the Abel-ergodic projection onto the kernel of $A$. The condition $0 \in ρ(A_0)$ is the natural abstract substitute for a spectral gap, and is sharp already in the Hilbertian case. We also obtain a companion divergence inequality. The arguments are remarkably short, yet the principle is genuinely unifying: it covers commutative and noncommutative situations on the same footing and can be used with arbitrary Banach spaces. As a consequence, we recover Poincaré inequalities associated with hypercontractive semigroups and extend their validity to semigroups satisfying only a reduced spectral gap and a reverse Riesz estimate. We then illustrate the flexibility of the method across a wide spectrum of geometries, ranging from Riemannian manifolds, Lie groups, metric measure spaces, spin manifolds to genuinely noncommutative settings such as group von Neumann algebras, semigroups of Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups and quantum tori, where we sometimes establish new inequalities and otherwise recover classical ones from a single principle.

math.FA↗
arXiv2026

Linear maps preserving families of C-symmetric operators

Let $\cH$ be a separable complex Hilbert space with $\dim\cH\ge3$. We characterize the bounded bijective complex-linear maps $T$ on $\BH$ for which there exists a bijection $ψ$ of the set of conjugations,preserving commutativity in both directions, such that $T$ maps the space of $C$-symmetric operators onto the space of $ψ(C)$-symmetric operators for every conjugation $C$. We show that this condition is equivalent to the existence of a bijection $φ$ of the set of all orthonormal bases such that $T$ maps the set of all $(e_n)$-diagonal operators onto the set of all $φ((e_n))$-diagonal operators. We also characterize the corresponding preservers in dimension two using Pauli coordinates, without assuming the commutativity condition on $ψ$.

math.FA↗
arXiv2026

Sharp embeddings between quasi-Banach Besov spaces and shallow ReLU variation spaces

Let $\mathbb{D}$ be the ridge dictionary generated by $\operatorname{ReLU}^k$ on a bounded Lipschitz domain $Ω\subset\mathbb{R}^d$. We establish sharp embeddings between Besov spaces and the associated variation space $\mathcal L_1(\mathbb{D})$ in the quasi-Banach range $0 k+d/p$ for $1<q\le\infty$. A rescaled-bump construction shows that this smoothness threshold is sharp. Conversely, for $0<p<1$, \[ \mathcal L_1(\mathbb{D})\hookrightarrow B^{k+1}_{p,2}(Ω), \] and both the smoothness $k+1$ and the fine index $2$ are optimal. The forward embedding converts Besov regularity of functions and solutions of partial differential equations into quantitative finite-width approximation bounds and greedy-algorithm convergence guarantees for shallow $\operatorname{ReLU}^k$ networks. The proofs use Littlewood-Paley localization, Fourier-Radon representations, measure-valued derivatives, and vector-valued singular integrals.

math.FA↗