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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 well-known characterization of shift-invariant spaces in terms of the Fourier transform. 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.

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., Comment: 11 pages

We consider bounded operators A acting iteratively on a finite set of vectors {fi: i ∈ I} in a Hilbert space ℌ and address the problem of providing necessary and sufficient conditions for the collection of iterates {Anfi: i ∈ I, n = 0, 1, 2,...} to form a frame for the space 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.

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 atmospheric surface fields from observed data (CPC and TRMM) and 6 RCM simulations belonging to the CORDEX Project. The wavelet used for filtering was Haar wavelet, but a comparative analysis with Daubechies 4 wavelet indicated that meteorological fields or regional indices were not very sensitive to the wavelet selected. Once the longer wavelengths were filtered, we focused on analyzing the spatial variability of extreme rainfall and the spatiotemporal variability of maximum and minimum surface air temperature on a daily basis. The results obtained suggest essential differences in the spatial distribution of the small-scale signal of extreme precipitation between TRMM and regional models, together with a large dispersion between models. While TRMM and CPC register a large signal throughout the continent, the RCMs place it over the Andes Cordillera and some over tropical South America. PAV signal for surface air temperature was found over the Andes Cordillera and the Brazilian Highlands, which are regions characterized by complex topography, and also on the coasts of the continent. The signal came specially from the small-scale stationary component. The transient part is much smaller than the stationary one, except over la Plata Basin where they are of the same order of magnitude. Also, the RCMs and CPC showed a large spread between them in representing this transient variability. The results confirm that RCMs have the potential to add value in the representation of extreme precipitation and the mean surface temperature in South America. However, this condition is not applicable throughout the whole continent but is particularly relevant in those terrestrial regions where the surface forcing is strong, such as the Andes Cordillera or the coasts of the continent.

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.

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 $$\begin{aligned} \{ A^n a^1, \ldots , A^n a^m \}_{n\ge 0} \end{aligned}$$ 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.

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 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.

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 $$\Gamma $$ 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 $$\Gamma $$ preserving. i.e. they commute with the action of $$\Gamma $$ . 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.

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.

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 $V$., 18 pages. To appear in Appl. Comput. Harmon. Anal

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.

In this note we solve the dynamical sampling problem for a class of shift-preserving (SP) operators, acting on a finitely generated shift-invariant space (FSIS). We find conditions on the operator and on a finite set of functions in the FSIS in order that the iterations of the operator on the functions produce a frame generator set. That is, the integer translations of the frame generator set is a frame of the FSIS. In order to obtain these results, we study the structure of SP operators and obtain a generalized finite dimensional spectral theorem.

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 Rⁿ 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., Facultad de Ciencias Exactas

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., Comment: 27 pages. To appear in Advances in Mathematics

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., Comment: 15 pages. Some typos corrected

In this note, we investigate the existence of frames of exponentials for L2(Ω) in the setting of LCA groups. Our main result shows that sub-multitiling properties of Ω⊂Gˆ with respect to a uniform lattice Γ of Gˆ 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. Fil: Barbieri, Davide. Universidad Autónoma de Madrid; España Fil: Cabrelli, Carlos. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Hernandez, Eugenio. Universidad Autónoma de Madrid; España Fil: Luthy, Peter. College Of Mount Saint Vincent; Estados Unidos Fil: Molter, Ursula Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Mosquera, Carolina Alejandra. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina

We establish some Balian-Low type results for Gabor spaces, which concern with discontinuity in some periodization of Z φ , where Z φ is the Zak transform of the window function φ ∈ L2 (ℝd). The results are compared with the case for shift-invariant spaces where the Zak transform is replaced by Fourier transform.

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,Anf i 〉, where h is the signal, (f i ) a system of fixed vectors and A a linear operator which is connected with the dynamical system. Here, we characterize those systems (Anf i ) n∈ℕ, i∈I with finite index sets I and normal operators A which are frames for the underlying Hilbert space. It turns out that this problem is intimately connected with spectral properties of the operator A and complex analysis in the unit disk. We also provide conditions on the spectral properties of A* for non-normal A.

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., 14 pages, 3 figures

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., 14 pages, 0 figures

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. Fil: Cabrelli, Carlos. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Molter, Ursula Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Pfander, Götz E.. Universitat Bremen. School of Enigineerring and Science Jacobs; Alemania

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. © 2009 Elsevier Inc. All rights reserved. Fil:Cabrelli, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Paternostro, V. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

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 ℝN and to infinite dimensional shift-invariant spaces in L2(ℝd). 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.

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 sampling problem in general shift invariant spaces. We characterize the functions in these spaces and provide necessary and sufficient conditions for a function in $L^2(\R)$ to belong to a sampling space. Furthermore we obtain decompositions of a sampling space in sampling subspaces. These decompositions are related with determining sets. Some examples are provided., Comment: 20 pages, to appear in "The Journal of Sampling Theory in Signal and Image Processing", typos corrected

Let φ : ℝd → ℂ be a compactly supported function which satisfies a refinement equation of the form [Formula: see text] where Γ ⊂ ℝd is a lattice, Λ is a finite subset of Γ, and A is a dilation matrix. We prove, under the hypothesis of linear independence of the Γ-translates of φ, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of L = [cAi-j]i,j∈Γ and a finite-dimensional subspace [Formula: see text] in the shift-invariant space generated by φ. We provide a basis of [Formula: see text] and show that its elements satisfy a property of homogeneity associated to the eigenvalues of L. If the function φ has accuracy κ, this basis can be chosen to contain a basis for all the multivariate polynomials of degree less than κ. These latter functions are associated to eigenvalues that are powers of the eigenvalues of A-1. Furthermore we show that the dimension of [Formula: see text] coincides with the local dimension of φ, and hence, every function in the shift-invariant space generated by φ can be written locally as a linear combination of translates of the homogeneous functions.

In this article, we develop a general method for constructing wavelets { | det A j | 1 / 2 ψ ( A j x − x j , k ) : j ∈ J , k ∈ K } on irregular lattices of the form X = { x j , k ∈ R d : j ∈ J , k ∈ K } , and with an arbitrary countable family of invertible d × d matrices { A j ∈ G L d ( R ) : j ∈ 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 nonharmonic Gabor frames with nonuniform 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.

In this paper we prove that if a Cantor set has ratios of dissection bounded away from zero, then there is a natural number N, such that its N-fold sum is an interval. Moreover, for each element z of this interval, we explicitly construct the N elements of C whose sum yields z. We also extend a result of Mendes and Oliveria showing that when s is irrational is an interval if and only if a /(1−2a) as/(1−2as) ≥ 1.

For a q q -regular Multiresolution Analysis of multiplicity r r with arbitrary dilation matrix A A for a general lattice Γ \Gamma in R n \mathbb {R}^n , we give necessary and sufficient conditions in terms of the mask and the symbol of the vector scaling function in order that an associated wavelet basis exists. We also show that if 2 r ( m − 1 ) ≥ n 2r(m-1) \geq n where m m is the absolute value of the determinant of A A , then these conditions are always met, and therefore an associated wavelet basis of q q -regular functions always exists. This extends known results to the case of multiwavelets in several variables with an arbitrary dilation matrix A A for a lattice Γ \Gamma .

Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σk∈Λ ck f(Ax−k), where Λ is a finite subset of Γ, the ck are r×r matrices, and f=(f1,...,fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)

We prove the existence of L pfunctions satisfying a kind of self-similarity condition. This is achieved by solving a functional equation by means of the construction of a contractive operator on an appropriate functional space. The solution, a fixed point of the operator, can be obtained by an iterative process, making this model very suitable to use in applications such as fractal image and signal compression. On the other hand, this “generalized self-similarity equation” includes matrix refinement equations of the typef(x) = ∑ ckf(Ax − k) which are central in the construction of wavelets and multiwavelets. The results of this paper will therefore yield conditions for the existence of L p-refinable functions in a very general setting.

We find conditions on the ratios of dissection of a Cantor set so that the group it generates under addition has positive Lebesgue measure. In particular, we answer affirmatively a special case of a conjecture posed by J. Palis.

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., 13 pages, To appear in J. Math. Anal. Appl

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, Comment: 23 pages

In this article we study the dilation equation f(x) = ∑h ch f (2x − h) in ℒ2(R) using a wavelet approach. We see that the structure of Multiresolution Analysis adapts very well to the study of scaling functions. The equation is reduced to an equation in a subspace of ℒ2(R) of much lower resolution. This simpler equation is then “wavelet transformed” to obtain a discrete dilation equation. In particular we study the case of compactly supported solutions and we see that conditions for the existence of solutions are given by convergence of infinite products of matrices. These matrices are of the type obtained by Daubechies, and, when the analyzing wavelet is the Haar wavelet, they are exactly the same.

In this paper we show that there exists an analytic expression for the Kantorovich distance between probebility measures on the circle. Previously such an expression was only known for measures supported on the real line. In the case that the measures are discrete, this formula enables us to show that the Kantorovich distance can be computed in linear time. This is important for applications, in particular in pattern recognition where this distance is used for texture analysis. As another application we see that the analytic expression found allows us to solve a Minimal Matching Problem in linear time, for which so far only n log n algorithms were known.

In this chapter 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 μ for which the set has positive and finite μ-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 0 or non-σ-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.

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 characterization of shift-modulation invariant spaces in this general context when $K$ and $\Lambda$ 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., Comment: 17 pages

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., 15 pages. Some corrections were added, in particular the title was changed. It will appear in "Sampling Theory in Signal and Image Processing"

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., 10 pages, revised version. To appear in Proceedings of the AMS.

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., Comment: 15 pages

Images with grey or colour levels admit a natural representation in terms of fuzzy sets, but without the usual probabilistic interpretation of the latter. We introduce a fuzzy set approach which incorporates, in part, the technique of iterated function systems (IFS) for the construction, analysis, and/or approximation of typically fractal sets and images. The method represents a significant departure from IFS, especially in the interpretation of the resulting image. The introduction of “grey-level maps,” ϑ i : [0, 1] → [0, 1] associated with the contractive maps w i of the IFS affords much greater flexibility in the generation of images as well as in the inverse problem.

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., 11 pages

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., 20 pages. A few corrections were added. To appear in "Sampling Theory in Signal and Image Processing"