arXiv subjects

Andrei Caragea

Publications and source records attributed to Andrei Caragea.

9 recordsLinked to original sources

arXiv2026

Schwartz-Class Gabor Windows and the Balian-Low Classification

Lattice Gabor frames consist of a window function and its time--frequency shifts by the elements of a lattice. This paper addresses the decay and smoothness achievable by a Gabor-frame window for a prescribed time--frequency/phase-space lattice, and gives a classification apart from the borderline cases specified below. Central to the paper is the introduction of an integer lattice parameter, the symplectic index gap. It governs the availability of regular, well-localized windows for a given lattice. For example, we show that a Schwartz-class Gabor-frame window exists exactly when this gap is at least the signal-space dimension, with the extended gap convention for symplectically irrational lattices. We formulate symplectic index gap Balian--Low theorems that place the classical and amalgam Balian--Low theorems in a common framework using the scale of modulation spaces. For symplectically rational lattices---which include those spanned by vectors with rational entries and which are the ones most relevant in applications---the necessary conditions for modulation-space regularity are also sufficient, except at the remaining lower-gap infinity endpoints. The companion paper "Tilings, Packings, and Smooth and Compactly Supported Gabor Windows" proves the associated geometric Tiling--$\varepsilon$--Packing theorem and constructs smooth, compactly supported Gabor windows.

math.FA↗

arXiv2026

Tilings, packings, and the existence of Schwartz-class Gabor windows

The existence and construction of Schwartz-class windows for lattice Gabor frames is a central problem in time--frequency analysis. For general time--frequency lattices, we characterize exactly those lattices that admit Schwartz-class windows and, additionally, windows in the Feichtinger algebra. Thereby we arrive at sharp classical and amalgam Balian--Low theorems for lattices in $\mathbb{R}^d$. For separable lattices and specified block-zero forms, we construct smooth windows that are compactly supported either in time or in frequency. These constructions are based on a characterization of pairs of lattices for which there exists a single set that tiles with one lattice and whose $\varepsilon$-neighborhood packs with the other.

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arXiv2025

Principal minors of Fourier matrices of square-free order

Chebotarev's theorem on roots of unity states that all minors of a Fourier matrix are non-zero if and only if the order of the matrix is prime. We establish cases in which all principal minors of Fourier matrices of square-free order are non-zero. In a subsequent paper we discuss the case of composites containing squares.

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arXiv2025

Minkowski chirality: a measure of reflectional asymmetry of convex bodies

Using an optimal containment approach, we quantify the asymmetry of convex bodies in $\mathbb{R}^n$ with respect to reflections across affine subspaces of a given dimension. We prove general inequalities relating these ''Minkowski chirality'' measures to Banach--Mazur distances and to each other, and prove their continuity with respect to the Hausdorff distance. In the planar case, we determine the reflection axes at which the Minkowski chirality of triangles and parallelograms is attained, and show that $\sqrt{2}$ is a tight upper bound on the chirality in both cases.

math.MG↗

arXiv2024

On the principal minors of Fourier matrices

For the $N$-dimensional Fourier matrix $\mathcal{F}_N$, we prove that if $N\geq 4$ is square-free, then every $2 \times 2$ and $3\times 3$ principal minor of $\mathcal{F}_N$ is nonzero. We also show that if $N\geq 4$ is not square-free, then $\mathcal{F}_N$ has zero principal minors of all sizes. Moreover, based on numerical experiments, we conjecture that if $N$ is square-free, then all principal minors of $\mathcal{F}_N$ are nonzero.

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arXiv2022

Time-Frequency Shift Invariance of Gabor Spaces with an $S_0$-Generator

We consider Gabor Riesz sequences generated by a lattice $Λ\subset \mathbb{R}^2$ and a window function $g \in L^2(\mathbb{R})$ which is well localized in both time and frequency. When $g$ belongs to the Feichtinger algebra, we prove that only those time-frequency shifts with parameters from the lattice $Λ$ leave the corresponding Gabor space invariant. This improves on earlier results where only lattices of rational density were considered. A slightly weaker result is proved - again for lattices of general density - under the regularity assumptions of the classical Balian-Low theorem, where both $g$ and its Fourier transform belong to the Sobolev space $H^1(\mathbb{R})$. The proof relies on a combination of methods from time-frequency analysis and the theory of $C^\ast$-algebras, specifically the so-called irrational rotation algebra.

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arXiv2022

Neural network approximation and estimation of classifiers with classification boundary in a Barron class

We prove bounds for the approximation and estimation of certain binary classification functions using ReLU neural networks. Our estimation bounds provide a priori performance guarantees for empirical risk minimization using networks of a suitable size, depending on the number of training samples available. The obtained approximation and estimation rates are independent of the dimension of the input, showing that the curse of dimensionality can be overcome in this setting; in fact, the input dimension only enters in the form of a polynomial factor. Regarding the regularity of the target classification function, we assume the interfaces between the different classes to be locally of Barron-type. We complement our results by studying the relations between various Barron-type spaces that have been proposed in the literature. These spaces differ substantially more from each other than the current literature suggests.

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arXiv2021

A note on exponential Riesz bases

We prove that if $I_\ell = [a_\ell,b_\ell)$, $\ell=1, \ldots, L$, are disjoint intervals in $[0,1)$ with the property that the numbers $1, a_1, \ldots, a_L, b_1, \ldots, b_L$ are linearly independent over $\mathbb{Q}$, then there exist pairwise disjoint sets $Λ_\ell \subset \mathbb{Z}$, $\ell=1, \ldots, L$, such that for every $J \subset \{ 1, \ldots , L \}$, the system $\{e^{2πi λx} : λ\in \cup_{\ell \in J} \, Λ_\ell \}$ is a Riesz basis for $L^2 ( \cup_{\ell \in J} \, I_\ell)$. Also, we show that for any disjoint intervals $I_\ell$, $\ell=1, \ldots, L$, contained in $[1,N)$ with $N \in \mathbb{N}$, the orthonormal basis $\{e^{2πi n x} : n \in \mathbb{Z} \}$ of $L^2[0,1)$ can be complemented by a Riesz basis $\{e^{2πi λx} : λ\inΛ\}$ for $L^2(\cup_{\ell=1}^L \, I_{\ell})$ with some set $Λ\subset (\frac{1}{N} \mathbb{Z}) \backslash \mathbb{Z}$, in the sense that their union $\{e^{2πi λx} : λ\in \mathbb{Z} \cup Λ\}$ is a Riesz basis for $L^2 ( [0,1) \cup I_1 \cup \cdots \cup I_L )$.

math.CA↗

arXiv2021

A quantitative subspace Balian-Low theorem

Let $\mathcal G\subset L^2(\mathbb R)$ be the subspace spanned by a Gabor Riesz sequence $(g,Λ)$ with $g\in L^2(\mathbb R)$ and a lattice $Λ\subset\mathbb R^2$ of rational density. It was shown recently that if $g$ is well-localized both in time and frequency, then $\mathcal G$ cannot contain any time-frequency shift $π(z) g$ of $g$ with $z\notinΛ$. In this paper, we improve the result to the quantitative statement that the $L^2$-distance of $π(z)g$ to the space $\mathcal G$ is equivalent to the Euclidean distance of $z$ to the lattice $Λ$, in the sense that the ratio between those two distances is uniformly bounded above and below by positive constants. On the way, we prove several results of independent interest, one of them being closely related to the so-called weak Balian-Low theorem for subspaces.

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