# Carlos CabrelliUniversity of Buenos Aires and CONICET (ARGENTINA) · Dept. of Math. and IMAS-UBA_CONICET

Carlos Cabrelli

Ph.D.

## About

115

Publications

16,253

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

1,633

Citations

Citations since 2016

Introduction

Additional affiliations

March 1993 - present

**CONICET-IMAS-UBA**

Position

- Investigador Superior

## Publications

Publications (115)

In this article we extend the theory of shift-invariant spaces to the context of LCA groups. We introduce the notion of H-invariant space for a countable discrete subgroup H of an LCA group G, and show that the concept of range function and the techniques of fiberization are valid in this context. As a consequence of this generalization we prove ch...

We prove the existence of sampling sets and interpolation sets near the
critical density, in Paley Wiener spaces of a locally compact abelian (LCA)
group G . This solves a problem left by Gr\"ochenig, Kutyniok, and Seip in the
article: `Landau's density conditions for LCA groups ' (J. of Funct. Anal. 255
(2008) 1831-1850). To achieve this result, w...

Complex-valued functions $f_1,\dots,f_r$ on ${\bold R}^d$ are
{\it refinable} if they are linear combinations of finitely many of the
rescaled and translated functions $f_i(Ax-k)$, where the translates $k$
are taken along a lattice $\Gamma \subset {\bold R}^d$ and $A$ is a
{\it dilation matrix} that expansively maps $\Gamma$ into itself.
Refinable...

Let $A$ be a dilation matrix, an $n \times n$ expansive matrix
that maps a full-rank lattice $\Gamma \subset {\bold R}^n$ into itself.
Let $\Lambda$ be a finite subset of $\Gamma$, and for $k \in \Lambda$
let $c_k$ be $r \times r$ complex matrices.
The refinement equation corresponding to $A$, $\Gamma$, $\Lambda$, and
$c = \{c_k\}_{k \in \Lambda}$...

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 Hilbert space $\mathcal{H}$. We completely characterize operators $T$ and $L$ and sets...

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...

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...

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 a...

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 \((...

In this paper, we prove the existence of a particular diagonalization for normal bounded operators defined on subspaces of L2(S) where 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 S with a discrete group of automorphisms. T...

A correction to this paper has been published: https://doi.org/10.1007/s43670-021-00010-6

In this note, we solve the dynamical sampling problem for a class of shift-preserving operators L:V→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 translation...

In this paper, we consider systems of vectors in a Hilbert space \(\mathcal {H}\) of the form \(\{g_{jk}: j \in J, \, k\in K\}\subseteq \mathcal {H}\), 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...

We characterize the normal operators A on ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell ^2$$\end{document} and the elements ai∈ℓ2\documentclass[12pt]{minimal}...

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 approx...

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

In this paper, we prove the existence of a particular diagonalization for normal bounded operators defined on subspaces of $L^2({\mathbf{R}})$ where ${\mathbf{R}}$ is a second countable LCA group. The subspaces where the operators act are invariant under the action of a group $\Gamma$ which is a semi-direct product of a uniform lattice of R with a...

In this article we study the structure of Γ-invariant spaces of L2(S). Here S 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 S and a group of automorphisms. This class includes in particular most of the crystallograph...

This paper aims to identify those regions within the South American continent where the Regional Climate Models (RCMs) have the potential to add value (PAV) compared to their coarser-resolution global forcing. For this, we used a spatial-scale filtering method based on the wavelet theory to distinguish the regional climatic signal present in atmosp...

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

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 approx...

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,\sig...

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 inte...

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 a...

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 partic...

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 the...

We characterize the normal operators $A$ on $\ell^2$ and the elements $a^i \in \ell^2$, with $1\le i\le m$, so that the sequence $\{ A^n a^1 , \ldots , A^n a^m \}_{n\ge 0}$ is Bessel or a frame. The characterization is given in terms of the backward shift invariant subspaces in $H^2(D)$.

In this article we study the structure of $\Gamma$-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 $\Gamma$, 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 part...

This contributed volume collects papers based on courses and talks given at the 2017 CIMPA school Harmonic Analysis, Geometric Measure Theory and Applications, which took place at the University of Buenos Aires in August 2017. These articles highlight recent breakthroughs in both harmonic analysis and geometric measure theory, particularly focusing...

In this note we investigate the existence of frames of exponentials for $L^2(\Omega)$ in the setting of LCA groups. Our main result shows that sub-multitiling properties of $\Omega \subset \widehat{G}$ with respect to a uniform lattice $\Gamma$ of $\widehat{G}$ guarantee the existence of a frame of exponentials with frequencies in a finite number o...

Dynamical Sampling aims to subsample solutions of linear dynamical systems at various times. One way to model this consists of considering inner products of the form h, A n fi, where h is the signal, (fi) a system of fixed vectors and A a linear operator which is connected with the dynamical system. Here, we characterize those systems (A n fi) n∈N,...

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. Some extreme cases are also considered. The result obtained shows a close analogy with the extra t...

Dynamical Sampling aims to sample spatio-temporal signals at fixed points in space and at various times, thereby exploiting the knowledge on the time evolution of the signal. In the corresponding mathematical model the samples of the signal $f$ are of the form $\langle f,A^nf_i\rangle$, where $A$ is a bounded operator and the $f_i$ are fixed vector...

We prove the existence of Riesz bases of exponentials of L 2 ( Ω ) L^2(\Omega ) , provided that Ω ⊂ R d \Omega \subset \mathbb {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 d...

Let H be Hilbert space and (Ω, m) a σ-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of L²(Ω, H) that are invariant under point-wise multiplication by functions from a fixed subset of L∞(Ω). Given a finite set of data F ⊆ L² (Ω, H), in this paper we prove the existence and construct an MI space M that best fits F,...

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 th...

Let $\mathcal{H}$ be Hilbert space and $(\Omega,\mu)$ a $\sigma$-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of $ L^2(\Omega, \mathcal{H})$ that are invariant under point-wise multiplication by functions in a fix subset of $L^{\infty}(\Omega).$ Given a finite set of data $\mathcal{F}\subseteq L^2(\Omega, \mathc...

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 proble...

This volume is a selection of written notes corresponding to courses taught at the CIMPA School: "New Trends in Applied Harmonic Analysis: Sparse Representations, Compressed Sensing and Multifractal Analysis". New interactions between harmonic analysis and signal and image processing have seen striking development in the last 10 years, and several...

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...

Let Y ={f(i),Af(i),...,Alif(i):i∈Ω},whereA is a bounded operator on l2(I). The problem under consideration is to find necessary and sufficient conditions on A, Ω, {li : i ∈ Ω} in order to recover any f ∈ l2(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 samp...

A Gabor space is a space generated by a discrete set of time-frequency shifted copies of a single window function. Starting from the question of whether a Gabor space contains additional time-frequency shifts of the window function we establish a new Balian-Low type result. This result extends (for example) the well established Amalgam Balian-Low T...

Given an arbitrary finite set of data F= {f_1,..., f_m} in L^2(R^d) 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 L^2(R^d). The
approximating subspace is required to have extra-invariance properties, that is
to be invariant under translation...

We consider smoothness properties of the generator of a principal Gabor space
on the real line which is invariant under some additional
translation-modulation pair. We prove that if a Gabor system on a lattice with
rational density is a Riesz basis for its closed linear span, and if the closed
linear span, a Gabor space, has any additional translat...

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...

A finitely generated shift invariant space V is a closed subspace of L2(Rd) that is generated by the integer translates of a finite number of functions. A set of frame generators for V is a set of functions whose integer translates form a frame for V . In this note we give necessary and sufficient conditions in order that a minimal set of frame gen...

In this article we study for which Cantor sets there exists a gauge-function
h, such that the h-Hausdorff-measure is positive and finite. We show that the
collection of sets for which this is true is dense in the set of all compact
subsets of a Polish space X. More general, any generic Cantor set satisfies
that there exists a translation-invariant...

A $(K,\Lambda)$ shift-modulation invariant space is a subspace of $L^2(G)$,
that is invariant by translations along elements in $K$ and modulations by
elements in $\Lambda$. Here $G$ is a locally compact abelian group, and $K$ and
$\Lambda$ are closed subgroups of $G$ and the dual group $\hat G$,
respectively. In this article we provide a character...

In this article we construct affine systems that provide a simultaneous
atomic decomposition for a wide class of functional spaces including the
Lebesgue spaces $L^p(\Rdst)$, $1<p<+\infty$. The novelty and difficulty of this
construction is that we allow for non-lattice translations.
We prove that for an arbitrary expansive matrix $A$ and any set $...

This article generalizes recent results in the extra invariance for shift-invariant spaces to the context of LCA groups. Let G be a locally compact abelian (LCA) group and K a closed subgroup of G. A closed subspace of L2(G) is called K-invariant if it is invariant under translations by elements of K. Assume now that H is a countable uniform lattic...

Given a set of points \F in a high dimensional space, the problem of finding
a union of subspaces \cup_i V_i\subset \R^N that best explains the data \F
increases dramatically with the dimension of \R^N. In this article, we study a
class of transformations that map the problem into another one in lower
dimension. We use the best model in the low dim...

In this note we study frame-related properties of a sequence of functions
multiplied by another function. In particular we study frame and Riesz basis
properties. We apply these results to sets of irregular translates of a
bandlimited function $h$ in $L^2(\R^d)$. This is achieved by looking at a set
of exponentials restricted to a set $E \subset \R...

A shift-invariant space is a space of functions that is invariant under integer translations. Such spaces are often used as models for spaces of signals and images in mathematical and engineering applications. This paper characterizes those shift-invariant subspaces S that are also invariant under additional (non-integer) translations. For the case...

This paper is a survey about recent results on sparse representa- tions and optimal models in dierent settings. Given a set of functions, we show that there exists an optimal collection of subspaces minimizing the sum of the square of the distances between each function and its closest subspace in the collection. Further, this collection of subspac...

In this article we study invariance properties of shift-invariant spaces in higher dimensions. We state and prove several necessary and sufficient conditions for a shift-invariant space to be invariant under a given closed subgroup of $\R^d$, and prove the existence of shift-invariant spaces that are exactly invariant for each given subgroup. As an...

In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovitch and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-Packing measures, for the family of dimension functions h, and characterize this classification in terms of the underlying sequences. Comment: 10 pages, revised version. To app...

A new paradigm in Sampling theory has been developed recently by Lu and Do. In this new approach the classical linear model is replaced by a non-linear, but structured model consisting of a union of subspaces. This is the natural approach for the new theory of compressed sampling, representation of sparse signals and signals with finite rate of inn...

Given a set of functions F={f1,…,fm}⊂L2(Rd), we study the problem of finding the shift-invariant space V with n generators {φ1,…,φn} that is “closest” to the functions of F in the sense that where wis are positive weights, and Vn is the set of all shift-invariant spaces that can be generated by n or less generators. The Eckart–Young theorem uses th...

Given a set of vectors (the data) in a Hilbert space ℋ, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for the given data, in a sense defined in the paper,...

In this article, we develop a general method for constructing wavelets
{|det A_j|^{1/2} g(A_jx-x_{j,k}): j in J, k in K}, on irregular lattices
of the form X={x_{j,k} in R^d: j in J, k in K}, and with an arbitrary
countable family of invertible dxd matrices {A_j in GL_d(R): j in J}
that do not necessarily have a group structure. This wavelet
constr...

Short Course on Wavelets and frames

In this chapter we discuss the problem of finding the shift-invariant space model that best fits a given class of observed
data F. If the data is known to belong to a fixed—but unknown—shift-invariant space V(Φ) generated by a vector function Φ, then we can probe the data F to find out whether the data is sufficiently rich for determining the shift...

Let $\phi: \R^d \longrightarrow \C$ be a compactly supported function which satisfies a refinement equation of the form $\phi(x) = \sum_{k\in\Lambda} c_k \phi(Ax - k),\quad c_k\in\C$, where $\Gamma\subset\R^d$ is a lattice, $\Lambda$ is a finite subset of $\Gamma$, and $A$ is a dilation matrix. We prove, under the hypothesis of linear independence...

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular, these functions include all the homogeneous polynomials that are reproducible by the generator, which links this representation to the accuracy of the space. We compl...

In this paper we present an overview of how geometric methods can be successfully used to solve problems in Analysis. We will focus on self- similar objects and use their structure to construct frames, Riesz bases and wavelet bases in Rd with a single generator function. Further, we show that the generating functions for these systems are dense in...

Given a function ψ in \({\cal L}^2({\Bbb R}^d),\) the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions
\(\{|{\rm det}\, a|^{j/2} \psi(a^jx-\gamma): j \in {\Bbb Z}, \gamma \in {\Gamma}\}.\) In this paper we prove that the set of functions generating affine systems that are a...

We provide a new representation of a refinable shift invariant space with a compactly supported generator, in terms of functions with a special property of homogeneity. In particular these functions include all the homogeneous polynomials that are reproducible by the generator, what links this representation to the accuracy of the space. We complet...

Sampling theory in spaces other than the space of band-limited functions has recently received considerable attention. This is in part because the band-limitedness assumption is not very realistic in many applications. In addition, band-limited functions have very slow decay which translates in poor reconstruction. In this article we study the samp...

Journé’s Lemma [11] is a critical component of many questions related to the product BMO theory of S.-Y. Chang and R. Fefferman. This article presents several different variants of the Lemma, in two and higher parameters, some known, some implicit in the literature, and some new. 1. Introduction, Journé’s Lemma We begin the discussion in two dimens...

In this paper we analyze Cantor type sets constructed by the removal of open intervals whose lengths are the terms of the p-sequence, {k-p}k=1∞. We prove that these Cantor sets are s-sets, by providing sharp estimates of their Hausdorff measure and dimension. Sets of similar structure arise when studying the set of extremal points of the boundaries...

We estimate the Hausdorff measure and dimension of Can- tor sets in terms of a sequence given by the lengths of the bounded complementary intervals. The results provide the relation between the decay rate of this sequence and the di- mension of the associated Cantor set. It is well-known that not every Cantor set on the line is an s-set for some 0...