arXiv subjects

Carlos Cabrelli

Publications and source records attributed to Carlos Cabrelli.

At least 19 recordsLinked to original sources

arXiv2025

Dynamical Sampling: A Survey

Dynamical sampling refers to a class of problems in which space-time samples are taken from a signal evolving under an underlying dynamical system. The goal is to use these samples to recover relevant information about the system, such as the initial state, the evolution operator, or the sources and sinks driving the dynamics. These problems are tightly connected to frame theory, operator theory, functional analysis, and other foundational areas of mathematics; they also give rise to new theoretical questions and have applications across engineering and the sciences. This survey provides an overview of the theoretical underpinnings of dynamical sampling, summarizes recent results, and outlines directions for future work, including open problems and conjectures.

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arXiv2025

Submodules of $H^2(\mathbb{T}^2)$ and Frames by Pairs of Bounded Commuting Operators

Recent work in Dynamical Sampling has been centered on characterizing frames obtained by the orbit of a vector under a bounded operator. We prove a necessary and sufficient condition for a pair of bounded commuting operators on a separable infinite-dimensional Hilbert space to generate a frame by unilateral iterations on a single vector. Applying the theory on submodules of the Hardy module $H^2(\mathbb{T}^2)$, we characterize these frames in terms of their relation to the two-variable Jordan block on a certain quotient module and provide some properties of frames of this form.

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arXiv2024

Periodic Source Detection in Discrete Dynamical Systems via space-time sampling

In this paper, we examine a discrete dynamical system defined by x(n+1) = Ax(n)+ w(n), where x takes values in a Hilbert space H and w is a periodic source with values in a fixed closed subspace W of H. Our goal is to identify conditions on some spatial sampling system G = {gj: j in J} of H that enable stable recovery of the unknown source term w from space-time samples { : n >=0,j in J}. We provide necessary and sufficient conditions on G = {g_j }_{j in J} to ensure stable recovery of any w in W . Additionally, we explicitly construct an operator R, dependent on G, such that R{ }_n,j} = w.

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arXiv2024

Weaving Riesz Bases

This paper explores woven frames in separable Hilbert spaces with an initial focus on the finite-dimensional case. We begin by simplifying the problem to bases, for which we obtain a unique characterization. We establish a condition that is both necessary and sufficient for vector reconstruction, which applies to Fourier matrices. Furthermore, we show that these characterizations are still valid in the infinite-dimensional case, for Riesz bases. Finally, we obtain several results for weaving Riesz bases of translations.

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arXiv2023

Learning optimal smooth invariant subspaces for data approximation

In this article, we consider the problem of approximating a finite set of data (usually huge in applications) by invariant subspaces generated through a small set of smooth functions. The invariance is either by translations under a full-rank lattice or through the action of crystallographic groups. Smoothness is ensured by stipulating that the generators belong to a Paley-Wiener space, that is selected in an optimal way based on the characteristics of the given data. To complete our investigation, we analyze the fundamental role played by the lattice in the process of approximation.

math.OC↗

arXiv2023

Reducing and Invariant subspaces under two commuting shift operators

In this article, we characterize reducing and invariant subspaces of the space of square integrable functions defined in the unit circle and having values in some Hardy space with multiplicity. We consider subspaces that reduce the bilateral shift and at the same time are invariant under the unilateral shift acting locally. We also study subspaces that reduce both operators. The conditions obtained are of the type of the ones in Helson and Beurling-Lax-Halmos theorems on characterizations of the invariance for the bilateral and unilateral shift. The motivations for our study were inspired by recent results on Dynamical Sampling in shift-invariant spaces.

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arXiv2023

Frames of iterations and vector-valued model spaces

Let T be a bounded operator on a Hilbert space H, and F = {f_j: j in J} an at most countable set of vectors in H. In this note, we characterize the pairs {T, F} such that {T^n f: f in F, n in I} form a frame of H, for the cases of I = N_0 and I = Z. The characterization for unilateral iterations gives a similarity with the compression of the shift acting on model spaces of the Hardy space of analytic functions defined on the unit disk with values in $l^2(J). This generalizes recent work for iterations of a single function. In the case of bilateral iterations, the characterization is by the bilateral shift acting on doubly invariant subspaces of L^2(T,l^2(J)). Furthermore, we characterize the frames of iterations for vector-valued model operators when J is finite in terms of Toeplitz and multiplication operators in the unilateral and bilateral case, respectively. Finally, we study the problem of finding the minimal number of orbits that produce a frame in this context.

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arXiv2021

Diagonalization of Shift-Preserving Operators

In this note we study the structure of shift-preserving operators acting on a finitely generated shift-invariant space. We define a new notion of diagonalization for these operators, which we call s-diagonalization. We give necessary and sufficient conditions on a bounded shift-preserving operator in order to be s-diagonalizable. These conditions are in terms of its range operator. We also obtain a generalized Spectral Theorem for normal bounded shift-preserving operators.

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arXiv2021

The structure of group preserving operators

In this paper, we prove the existence of a particular diagonalization for normal bounded operators defined on subspaces of $L^2(\mathfrak{S})$ where $\mathfrak{S}$ is a second countable LCA group. The subspaces where the operators act are invariant under the action of a group $Γ$ which is a semi-direct product of a uniform lattice of $\mathfrak{S}$ with a discrete group of automorphisms. This class includes the crystal groups which are important in applications as models for images. The operators are assumed to be $Γ$ preserving. i.e. they commute with the action of $Γ$. In particular we obtain a spectral decomposition for these operators. This generalizes recent results on shift-preserving operators acting on lattice invariant subspaces where $\mathfrak{S}$ is the Euclidean space.

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arXiv2020

Multi-orbital frames through model spaces

We characterize the normal operators $A$ on $\ell^2$ and the elements $a^i \in \ell^2$, with $1\le i\le m$, such that the sequence $$\{ A^n a^1 , \ldots , A^n a^m \}_{n\ge 0}$$ is a frame. The characterization makes strong use of the pseudo-hyperbolic metric of $\mathbb{D}$ and is given in terms of the backward shift invariant subspaces of $H^2(\mathbb{D})$ associated to finite products of interpolating Blaschke products.

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arXiv2020

Approximation by group invariant subspaces

In this article we study the structure of $Γ$-invariant spaces of $L^2(\bf R)$. Here $\bf R$ is a second countable LCA group. The invariance is with respect to the action of $Γ$, a non commutative group in the form of a semidirect product of a discrete cocompact subgroup of $\bf R$ and a group of automorphisms. This class includes in particular most of the crystallographic groups. We obtain a complete characterization of $Γ$-invariant subspaces in terms of range functions associated to shift-invariant spaces. We also define a new notion of range function adapted to the $Γ$-invariance and construct Parseval frames of orbits of some elements in the subspace, under the group action. These results are then applied to prove the existence and construction of a $Γ$-invariant subspace that best approximates a set of functional data in $L^2(\bf R)$. This is very relevant in applications since in the euclidean case, $Γ$-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing.

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arXiv2020

Riesz bases of exponentials and the Bohr topology

We provide a necessary and sufficient condition to ensure that a multi-tile $Ω$ of $R^d$ of positive measure (but not necessarily bounded) admits a structured Riesz basis of exponentials for $ L^{2}(Ω)$. New examples are given and this characterization is generalized to abstract locally compact abelian groups.

math.CA↗

arXiv2019

Local-to-global frames and applications to dynamical sampling problem

In this paper we consider systems of vectors in a Hilbert space $\mathcal{H}$ of the form $\{g_{jk}: j \in J, \, k\in K\}\subset \mathcal{H}$ where $J$ and $K$ are countable sets of indices. We find conditions under which the local reconstruction properties of such a system extend to global stable recovery properties on the whole space. As a particular case, we obtain new local-to-global results for systems of type $\{A^ng\}_{g\in\mathcal{G},0\leq n\leq L }$ arising in the dynamical sampling problem.

math.FA↗

arXiv2019

Optimal translational-rotational invariant dictionaries for images

We provide the construction of a set of square matrices whose translates and rotates provide a Parseval frame that is optimal for approximating a given dataset of images. Our approach is based on abstract harmonic analysis techniques. Optimality is considered with respect to the quadratic error of approximation of the images in the dataset with their projection onto a linear subspace that is invariant under translations and rotations. In addition, we provide an elementary and fully self-contained proof of optimality, and the numerical results from datasets of natural images.

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arXiv2019

Subspaces with extra invariance nearest to observed data

Given an arbitrary finite set of data F= {f_1,..., f_m} in L2(Rd) we prove the existence and show how to construct a "small shift invariant space" that is "closest" to the data F over certain class of closed subspaces of L2(Rd). The approximating subspace is required to have extra-invariance properties, that is to be invariant under translations by a prefixed additive subgroup of Rd containing Zd. This is important for example in situations where we need to deal with jitter error of the data. Here small means that our solution subspace should be generated by the integer translates of a small number of generators. We give an expression for the error in terms of the data and construct a Parseval frame for the optimal space. We also consider the problem of approximating F from generalised Paley-Wiener spaces of Rd that are generated by the integer translates of a finite number of functions. That is finitely generated shift invariant spaces that are translation invariant. We characterise these spaces in terms of multi-tile sets of Rd, and show the connections with recent results on Riesz basis of exponentials on bounded sets of Rd. Finally we study the discrete case for our approximation problem.

math.FA↗

arXiv2018

A Fourier Frame for the Middle-Third Cantor Measure

In this paper we show that if $μ$ is any locally and uniformly $α$-dimensional measure supported on a $α$-quasi-regular set $E$, then $L^2(μ)$ admits a frame of exponentials. In particular, for the uniform middle third Cantor measure, $μ_C,$ our result shows that there exists a countable set $Λ$ such that $\{e^{2πi t λ}\}_{λ\in Λ}$ is a frame for $L^2(μ_C)$ (i.e. the measure $μ_C$ admits a generalized spectrum), answering an old outstanding question about the existence of a frame of exponentials for the space $L^2(μ_C)$.

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arXiv2017

Time-frequency shift invariance of Gabor spaces generated by integer lattices

We study extra time-frequency shift invariance properties of Gabor spaces. For a Gabor space generated by an integer lattice, we state and prove several characterizations for its time-frequency shift invariance with respect to a finer integer lattice. The extreme cases of full translation invariance, full modulation invariance, and full time-frequency shift invariance are also considered. The results show a close analogy with the extra translation invariance of shift-invariant spaces.

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arXiv2017

Riesz Bases of Exponentials on Unbounded Multi-tiles

We prove the existence of Riesz bases of exponentials of L^2(Omega), provided that Omega in R^d is a measurable set of finite and positive measure, not necessarily bounded, that satisfies a multi-tiling condition and an arithmetic property that we call admissibility. This property is satisfied for any bounded domain, so our results extend the known case of bounded multi-tiles. We also extend known results for submulti-tiles and frames of exponentials to the unbounded case.

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