Search

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.

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 contribution we establish a dictionary between terms in two different areas in mathematics: Dynamical Sampling and Control Theory. The purpose is to show that many of the topics studied in either field are also addressed in the other one, just with a different terminology. We further analyze the relations between discrete-time and continuous-time versions of the problem, using results from both fields.

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.

The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in $$\mathbb {R}^2$$ , provides a useful way to classify pointwise singularity. In this paper we characterize all functions whose 2-microlocal frontier at a given point $$x_0$$ is a given line. Further, for a general concave downward curve, we obtain a large family of functions (or distributions) for which the 2-microlocal frontier is the given curve. This family contains—as special cases—the constructions given in Meyer (CRM monograph series, 1998), Guiheneuf et al. (ACHA 5(4):487–492, 1998) and Levy Vehel et al. (Proc Symp Pure Math 72:319–334, 2004). Moreover, following Levy Vehel et al. (2004), we extend our results to the prescription on a countable dense set.

Given any dimension function $h$, we construct a perfect set $E \subseteq \mathbb{R}$ of zero $h$-Hausdorff measure, that contains any finite polynomial pattern. This is achieved as a special case of a more general construction in which we have a family of functions $\mathcal{F}$ that satisfy certain conditions and we construct a perfect set $E$ in $\mathbb{R}^N$, of $h$-Hausdorff measure zero, such that for any finite set $\{ f_1,\ldots,f_n\}\subseteq \mathcal{F}$, $E$ satisfies that $\bigcap_{i=1}^n f^{-1}_i(E)\neq\emptyset$. We also obtain an analogous result for the images of functions. Additionally we prove some related results for countable (not necessarily finite) intersections, obtaining, instead of a perfect set, an $\mathcal{F}_{\sigma}$ set without isolated points., Comment: To appear in Mathematical Proceedings of the Cambridge Philosophical Society

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.

Let $$\varGamma $$ be a crystal group in $$\mathbb {R}^d$$ . A function $$\varphi :\mathbb {R}^d\longrightarrow \mathbb {C}$$ is said to be crystal-refinable (or $$\varGamma $$ -refinable) if it is a linear combination of finitely many of the rescaled and translated functions $$\varphi (\gamma ^{-1}(ax))$$ , where the translations $$\gamma $$ are taken on a crystal group $$\varGamma $$ , and a is an expansive dilation matrix such that $$a\varGamma a^{-1}\subset \varGamma .$$ A $$\varGamma $$ -refinable function $$\varphi : \mathbb {R}^d \rightarrow \mathbb {C}$$ satisfies a refinement equation $$\varphi (x)=\sum _{\gamma \in \varGamma }d_\gamma \varphi (\gamma ^{-1}(ax))$$ with $$d_\gamma \in \mathbb {C}$$ . Let $$\mathcal S(\varphi )$$ be the linear span of $$\{\varphi (\gamma ^{-1}(x)): \gamma \in \varGamma \}$$ and $$\mathcal {S}^h=\{f(x/h):f\in \mathcal {S(\varphi )}\}$$ . One important property of $$\mathcal S(\varphi )$$ is, how well it approximates functions in $$L^2(\mathbb {R}^d)$$ . This property is very closely related to the crystal-accuracy of $$\mathcal S(\varphi )$$ , which is the highest degree p such that all multivariate polynomials q(x) of $$\mathrm{degree}(q)

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 paper we study the continuous dynamical sampling problem at infinite time in a complex Hilbert space H . We find necessary and sufficient conditions on a bounded linear operator A ∈ B ( H ) and a set of vectors G ⊂ H , in order to obtain that { e t A g } g ∈ G , t ∈ [ 0 , ∞ ) is a semi-continuous frame for H . We study if it is possible to discretize the time variable t and still have a frame for H . We also relate the continuous iteration e t A on a set G to the discrete iteration ( A ′ ) n on G ′ for an adequate operator A ′ and set G ′ ⊂ H .

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 consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {Fi : i ∈ N}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps Fi are of the form Fi(x) = rix + bi on X = R d , we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi ri is strictly smaller than 1. Further, if ρ = {ρk}k∈N is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set. Fil: Barrozo, María Fernanda. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; Argentina Fil: Molter, Ursula Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina

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

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.

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.

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.

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 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 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.

In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $\alpha,\beta\in(0,1]$, we will say that a set $E\subset \R^2$ is an $F_{\alpha\beta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $\beta$ and, for each direction $e$ in $L$, there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is equal or greater than $\alpha$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $\dim(E)\ge\max\left\{\alpha+\frac{\beta}{2} ; 2\alpha+\beta -1\right\}$ for any $E\in F_{\alpha\beta}$. In particular we are able to extend previously known results to the ``endpoint'' $\alpha=0$ case., Comment: 13 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

For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $\alpha$ or $F_\alpha$-set if for each direction $e$ in the unit circle there is a line segment $\ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap\ell_e$ is greater or equal than $\alpha$. In this paper we show that if $\alpha > 0$, there exists a set $E\in F_\alpha$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2\alpha}\log^{-\theta}(\frac{1}{x})$, $\theta>\frac{1+3\alpha}{2}$, which improves on the the previously known bound, that $H^{\beta}(E) = 0$ for $\beta>1/2+3/2\alpha$. Further, by refining the argument in a subtle way, we are able to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for $\h_\gamma(x)=\log^{-\gamma}(\frac{1}{x})$, $\gamma>0$, we construct a set $E_\gamma\in F_{\h_\gamma}$ of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any $E\in F_{\h_\gamma}$, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_\gamma$., Comment: Final version

In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of "zero dimensional" Furstenberg sets., 16 pages, 3 figures