arXiv subjects

Carlos Cabrelli

Publications and source records attributed to Carlos Cabrelli.

At least 37 records · Page 2Linked to original sources

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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arXiv2017

Frames of exponentials and sub-multitiles in LCA groups

In this note we investigate the existence of frames of exponentials for $L^2(Ω)$ in the setting of LCA groups. Our main result shows that sub-multitiling properties of $Ω\subset \widehat{G}$ with respect to a uniform lattice $Γ$ of $\widehat{G}$ guarantee the existence of a frame of exponentials with frequencies in a finite number of translates of the annihilator of $Γ$. We also prove the converse of this result and provide conditions for the existence of these frames. These conditions extend recent results on Riesz bases of exponentials and multitilings to frames.

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arXiv2016

An Approximation Problem in Multiplicatively Invariant Spaces

Let $\mathcal{H}$ be Hilbert space and $(Ω,μ)$ a $σ$-finite measure space. Multiplicatively invariant (MI) spaces are closed subspaces of $ L^2(Ω, \mathcal{H})$ that are invariant under point-wise multiplication by functions in a fix subset of $L^{\infty}(Ω).$ Given a finite set of data $\mathcal{F}\subseteq L^2(Ω, \mathcal{H}),$ in this paper we prove the existence and construct an MI space $M$ that best fits $\mathcal{F}$, in the least squares sense. MI spaces are related to shift invariant (SI) spaces via a fiberization map, which allows us to solve an approximation problem for SI spaces in the context of locally compact abelian groups. On the other hand, we introduce the notion of decomposable MI spaces (MI spaces that can be decomposed into an orthogonal sum of MI subspaces) and solve the approximation problem for the class of these spaces. Since SI spaces having extra invariance are in one-to-one relation to decomposable MI spaces, we also solve our approximation problem for this class of SI spaces. Finally we prove that translation invariant spaces are in correspondence with totally decomposable MI spaces.

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arXiv2015

Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups

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 Gröchenig, 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, we prove the existence of universal Riesz bases of characters for L2(Omega), provided that the relatively compact subset Omega of the dual group of G satisfies a multi-tiling condition. This last result generalizes Fuglede's theorem, and extends to LCA groups setting recent constructions of Riesz bases of exponentials in bounded sets of Rd.

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arXiv2014

Time-frequency shift invariance and the Amalgan Balian Low Theorem

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 translation-modulation invariance, then its generator cannot decay well in time and in frequency simultaneously.

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arXiv2013

Linear combinations of frame generators in systems of translates

A finitely generated shift invariant space $V$ is a closed subspace of $L^2(\R^d)$ 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 generators can be obtained by taking linear combinations of the given frame generators. Surprisingly the results are very different to the recently studied case when the property to be a frame is not required.

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arXiv2012

Non-uniform painless decompositions for anisotropic Besov and Triebel-Lizorkin spaces

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 $Λ$ - satisfying a certain spreadness condition but otherwise irregular- there exists a smooth window whose translations along the elements of $Λ$ and dilations by powers of $A$ provide an atomic decomposition for the whole range of the anisotropic Triebel-Lizorkin spaces. The generating window can be either chosen to be bandlimited or to have compact support. To derive these results we start with a known general "painless" construction that has recently appeared in the literature. We show that this construction extends to Besov and Triebel-Lizorkin spaces by providing adequate dual systems.

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arXiv2012

Shift-modulation invariant spaces on LCA groups

A $(K,Λ)$ 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 $Λ$. Here $G$ is a locally compact abelian group, and $K$ and $Λ$ are closed subgroups of $G$ and the dual group $\hat G$, respectively. In this article we provide a characterization of shift-modulation invariant spaces in this general context when $K$ and $Λ$ are uniform lattices. This extends previous results known for $L^2(\R^d)$. We develop fiberization techniques and suitable range functions adapted to LCA groups needed to provide the desired characterization.

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arXiv2011

Frames by Multiplication

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 $E \subset \R^d$ with frequencies in a countable set $Λ$ and multiplying it by the Fourier transform of a fixed function $h \in L^2(E)$. Using density results due to Beurling, we prove the existence and give ways to construct frames by irregular translates.

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arXiv2011

Visible and Invisible Cantor sets

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 measure mu for which the set has positive and finite mu-measure. In contrast, we generalize an example of Davies of dimensionless Cantor sets (i.e. a Cantor set for which any translation invariant measure is either zero or non-sigma-finite, that enables us to show that the collection of these sets is also dense in the set of all compact subsets of a Polish space X.

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arXiv2011

A Dimension Reduction Scheme for the Computation of Optimal Unions of Subspaces

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 dimensional space to approximate the best solution in the original high dimensional space. We then estimate the error produced between this solution and the optimal solution in the high dimensional space.

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arXiv2010

Invariance of a Shift-Invariant Space in Several Variables

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 application we relate the extra invariance to the size of support of the Fourier transform of the generators of the shift-invariant space. This work extends recent results obtained for the case of one variable to several variables.

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arXiv2009

Shift Invariant Spaces on LCA Groups

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 characterizations of frames and Riesz bases of these spaces extending previous results, that were known for Rd and the lattice Zd .

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arXiv2009

Sampling in a Union of Frame Generated Subspaces

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 innovation. In this article we extend the theory of Lu and Do, for the case that the subspaces in the union are shift-invariant spaces. We describe the subspaces by means of frame generators instead of orthonormal bases. We show that, the one to one and stability conditions for the sampling operator, are valid for this more general case.

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arXiv2009

Extra Invariance of Shift-Invariant Spaces on LCA Groups

Let G be an LCA group and K a closed subgroup of G. A closed subspace of L^2(G) is called K-invariant if it is invariant under translations by elements of K. Assume now that H is a countable uniform lattice in G and M is any closed subgroup of G containing H. In this article we study necessary and sufficient conditions for an H-invariant space to be M-invariant. As a consequence of our results we prove that for each closed subgroup M of G containing the lattice H, there exists an H-invariant space S that is exactly M-invariant. That is, S is not invariant under any other subgroup M' containing M. We also obtain estimates on the support of the Fourier transform of the generators of the H-invariant space, related to its M-invariance.

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arXiv2008

Invariance of a Shift-Invariant Space

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 of finitely generated spaces, these spaces are characterized in terms of the generators of the space. As a consequence, it is shown that principal shift-invariant spaces with a compactly supported generator cannot be invariant under any non-integer translations.

math.FA↗

arXiv2008

Optimal Non-Linear Models for Sparsity and Sampling

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, and provides an optimal model for sampling in union of subspaces. The results are proved in a general setting and then applied to the case of low dimensional subspaces of R^N and to infinite dimensional shift-invariant spaces in L^2(R^d). We also present an iterative search algorithm for finding the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary design, signal models for signals with finite rate of innovation, and the subspace segmentation problem.

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arXiv2007

Wavelets on Irregular Grids with Arbitrary Dilation Matrices, and Frames Atoms for L^2(R^d)

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 construction is a particular case of general atomic frame decompositions of L^2(R^d) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R^d. Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.

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