arXiv subjects

C. Cabrelli

Publications and source records attributed to C. Cabrelli.

9 recordsLinked to original sources

arXiv2025

Optimal Dynamical Frames

Motivated by the dynamical sampling problem, we study frames in an infinite dimensional Hilbert space generated by the iterates of a bounded operator T, also known as dynamical frames. We first characterize the operators that generate Parseval dynamical frames by showing that the previously known sufficient conditions for their existence are also necessary. We then introduce the frame index of T, the minimal number of vectors required to generate a frame by iterations, and derive an explicit formula for it in the Parseval case together with a general condition for the non-Parseval setting. Finally, we prove that if both T and T* admit frames of iterations, then their frame indices coincide through an explicit construction.

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arXiv2022

Frames by orbits of two operators that commute

Frames formed by orbits of vectors through the iteration of a bounded operator have recently attracted considerable attention, in particular due to its applications to dynamical sampling. In this article, we consider two commuting bounded operators acting on some separable Hilbert space $\mathcal H$. We completely characterize operators $T$ and $L$ with $TL=LT$ and sets $\Phi\subset \mathcal H$ such that the collection $\{T^k L^j \phi: k\in \mathbb Z, j\in J, \phi \in \Phi \}$ forms a frame of $\mathcal H$. This is done in terms of model subspaces of the space of square integrable functions defined on the torus and having values in some Hardy space with multiplicity. The operators acting on these models are the bilateral shift and the compression of the unilateral shift (acting pointwisely). This context includes the case when the Hilbert space $\mathcal H$ is a subspace of $L^2(\mathbb R)$, invariant under translations along the integers, where the operator $T$ is the translation by one and $L$ is a shift-preserving operator.

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arXiv2020

Data approximation with time-frequency invariant systems

In this paper we prove the existence of a time-frequency space that best approximates a given finite set of data. Here best approximation is in the least square sense, among all time-frequency spaces with no more than a prescribed number of generators. We provide a formula to construct the generators from the data and give the exact error of approximation. The setting is in the space of square integrable functions defined on a second countable LCA group and we use the Zak transform as the main tool.

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arXiv2019

Extra-invariance of group actions

Given discrete groups $\Gamma \subset \Delta$ we characterize $(\Gamma,\sigma)$-invariant spaces that are also invariant under $\Delta$. This will be done in terms of subspaces that we define using an appropriate Zak transform and a particular partition of the underlying group. On the way, we obtain a new characterization of principal $(\Gamma,\sigma)$-invariant spaces in terms of the Zak transform of its generator. This result is in the spirit of the analogous in the context of shift-invariant spaces in terms of the Fourier transform, which is very well-known. As a consequence of our results, we give a solution for the problem of finding the $(\Gamma,\sigma)$-invariant space nearest - in the sense of least squares - to a given set of data.

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arXiv2019

Dynamical Sampling for Shift-preserving Operators

In this note, we solve the dynamical sampling problem for a class of shift-preserving operators $L:V\to V$ acting on a finitely generated shift-invariant space $V$. We find conditions on $L$ and a finite set of functions of $V$ so that the iterations of the operator $L$ on the functions produce a frame generator set of $V$. This means that the integer translations of the generators form a frame of $V$.

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arXiv2017

Dynamical Sampling on Finite Index Sets

We consider bounded operators $A$ acting iteratively on a finite set of vectors $\{f_i : i\in I\}$ in a Hilbert space $\mathcal H$ and address the problem of providing necessary and sufficient conditions for the collection of iterates $\{A^nf_i : i\in I,\,n=0,1,2,\ldots\}$ to form a frame for the space $\mathcal H$. For normal operators $A$ we completely solve the problem by proving a characterization theorem. Our proof incorporates techniques from different areas of mathematics, such as operator theory, spectral theory, harmonic analysis, and complex analysis in the unit disk. In the second part of the paper we drop the strong condition on $A$ to be normal. Despite this quite general setting, we are able to prove a characterization which allows to infer many strong necessary conditions on the operator $A$. For example, $A$ needs to be similar to a contraction of a very special kind. We also prove a characterization theorem for the finite-dimensional case. --- These results provide a theoretical solution to the so-called Dynamical Sampling problem where a signal $f$ that is evolving in time through iterates of an operator $A$ is spatially sub-sampled at various times and one seeks to reconstruct the signal $f$ from these spatial-temporal samples.

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arXiv2016

Existence of quasicrystals and universal stable sampling and interpolation in LCA groups

We characterize all the locally compact abelian (LCA) groups that contain quasicrystals (a class of model sets). Moreover, we describe all possible quasicrystals in the group constructing an appropriate lattice associated with the cut and project scheme that produces it. On the other hand, if an LCA group G admits a simple quasicrystal, we prove that recent results of Meyer and Matei for the case of the n-dimensional Euclidean space can be extended to G. More precisely, we prove that simple quasicrystals are universal sets of stable sampling and universal sets of stable interpolation in generalized Paley-Wiener spaces.

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arXiv2016

Iterative actions of normal operators

Let $A$ be a normal operator in a Hilbert space $\mathcal{H}$, and let $\mathcal{G} \subset \mathcal{H}$ be a countable set of vectors. We investigate the relations between $A$, $\mathcal{G}$ , and $L$ that makes the system of iterations $\{A^ng: g\in \mathcal{G},\;0\leq n< L(g)\}$ complete, Bessel, a basis, or a frame for $\mathcal{H}$. The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis.

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arXiv2014

Dynamical sampling

Let Y={f(i), Af(i),..., A^{li} f(i): i in Omega}, where A is a bounded operator on l^2(I). The problem under consideration is to find necessary and sufficient conditions on A, Omega, {l_i:i in Omega} in order to recover any f \in l^2(I) from the measurements Y. This is the so called dynamical sampling problem in which we seek to recover a function f by combining coarse samples of f and its futures states A^l f. We completely solve this problem in finite dimensional spaces, and for a large class of self adjoint operators in infinite dimensional spaces. In the latter case, the Müntz-Szász Theorem combined with the Kadison-Singer/Feichtinger Theorem allows us to show that Y can never be a Riesz basis when Omega is finite. We can also show that, when Omega is finite, Y={f(i), Af(i),..., A^{li}f(i): i in Omega} is not a frame except for some very special cases. The existence of these special cases is derived from Carleson's Theorem for interpolating sequences in the Hardy space H^2(D).

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