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## Sparsity Equivalence of Anisotropic Decompositions

Kutyniok, Gitta
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
Anisotropic decompositions using representation systems such as curvelets, contourlet, or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of functions exhibiting singularities on lower dimensional embedded manifolds. The literature now contains various direct proofs of this fact and of related sparse approximation results. However, it seems quite cumbersome to prove such a canon of results for each system separately, while many of the systems exhibit certain similarities. In this paper, with the introduction of the concept of sparsity equivalence, we aim to provide a framework which allows categorization of the ability for sparse approximations of representation systems. This framework, in particular, enables transferring results on sparse approximations from one system to another. We demonstrate this concept for the example of curvelets and shearlets, and discuss how this viewpoint immediately leads to novel results for both systems.; Comment: 20 pages, 4 figures

## Discretization of continuous frame

Fattahi, A.; Javanshiri, H.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
In this paper we consider on the notion of continuous frame of subspace and define a new concept of continuous frame, entitled {\it continuous atomic resolution of identity}, for arbitrary Hilbert space $\h$ which has a countable reconstruction formula. Among the other result, we characterize the relationship between this new concept and other known continuous frame. Finally, we state and prove the assertions of the stability of perturbation in this concept.; Comment: 16, pages

## Recovery of Missing Samples in Oversampling Formulas for Band Limited Functions

Del Prete, Vincenza
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.4%
In a previous paper, the author constructed frames and oversampling formulas for band-limited functions, in the framework of the theory of shift-invariant spaces. In this article we study the problem of recovering missing samples. We find a sufficient condition for the recovery of a finite set of missing samples. The condition is expressed as a linear independence of the components of a vector W over the space of trigonometric polynomials determined by the frequencies of the missing samples. We apply the theory to the derivative sampling of any order and we illustrate our results with a numerical experiment.; Comment: 19 pages, 3 figures, corrected a few typos

## Gabor fusion frames generated by difference sets

Bojarovska, Irena; Paternostro, Victoria
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
Collections of time- and frequency-shifts of suitably chosen generators (Alltop or random vectors) proved successful for many applications in sparse recovery and related fields. It was shown in \cite{xia2005achieving} that taking a characteristic function of a difference set as a generator, and considering only the frequency shifts, gives an equaingular tight frame for the subspace they span. In this paper, we investigate the system of all $N^2$ time- and frequency-shifts of a difference set in dimension $N$ via the mutual coherence, and compare numerically its sparse recovery effectiveness with Alltop and random generators. We further view this Gabor system as a fusion frame, show that it is optimally sparse, and moreover an equidistant tight fusion frame, i.e. it is an optimal Grassmannian packing.; Comment: 14 pages, 3 figures

## Series expansions in Fr\'echet spaces and their duals; construction of Fr\'echet frames

Pilipović, Stevan; Stoeva, Diana T.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
Frames for Fr\'echet spaces $X_F$ with respect to Fr\'echet sequence spaces $\Theta_F$ are studied and conditions, implying series expansions in $X_F$ and $X_F^*$, are determined. If $\{g_i\}$ is a $\Theta_0$-frame for $X_0$, we construct a sequence $\{X_s\}_{s\in {\mathbb N}_0}$, $X_s\subset X_{s-1}$, $s\in {\mathbb N}$, for given $\Theta_F$, respectively a sequence $\{\Theta_s\}_{s\in {\mathbb N}_0}$, $\Theta_s\subset \Theta_{s-1}$, $s\in {\mathbb N}$, for given $X_F$, so that $\{g_i\}$ is a pre-$F$-frame (or $F$-frame) for $X_F$ with respect to $\Theta_F$ under different assumptions given on $X_0$, $\Theta_0$, $\Theta_F$ or $X_0$, $\Theta_0$, $X_F$.

## Shrinking and boundedly complete atomic decompositions in Fr\'echet spaces

Bonet, José; Fernández, Carmen; Galbis, Antonio; Ribera, Juan M.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
We study atomic decompositions in Fr\'echet spaces and their duals, as well as perturbation results. We define shrinking and boundedly complete atomic decompositions on a locally convex space, study the duality of these two concepts and their relation with the reflexivity of the space. We characterize when an unconditional atomic decomposition is shrinking or boundedly complete in terms of properties of the space. Several examples of concrete atomic decompositions in function spaces are also presented.

## Approximation of Fourier Integral Operators by Gabor multipliers

Cordero, Elena; Gröchenig, Karlheinz; Nicola, Fabio
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
A general principle says that the matrix of a Fourier integral operator with respect to wave packets is concentrated near the curve of propagation. We prove a precise version of this principle for Fourier integral operators with a smooth phase and a symbol in the Sjoestrand class and use Gabor frames as wave packets. The almost diagonalization of such Fourier integral operators suggests a specific approximation by (a sum of) elementary operators, namely modified Gabor multipliers. We derive error estimates for such approximations. The methods are taken from time-frequency analysis.; Comment: 22. pages

## Constructing all self-adjoint matrices with prescribed spectrum and diagonal

Fickus, Matthew; Mixon, Dustin G.; Poteet, Miriam J.; Strawn, Nate
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
The Schur-Horn Theorem states that there exists a self-adjoint matrix with a given spectrum and diagonal if and only if the spectrum majorizes the diagonal. Though the original proof of this result was nonconstructive, several constructive proofs have subsequently been found. Most of these constructive proofs rely on Givens rotations, and none have been shown to be able to produce every example of such a matrix. We introduce a new construction method that is able to do so. This method is based on recent advances in finite frame theory which show how to construct frames whose frame operator has a given prescribed spectrum and whose vectors have given prescribed lengths. This frame construction requires one to find a sequence of eigensteps, that is, a sequence of interlacing spectra that satisfy certain trace considerations. In this paper, we show how to explicitly construct every such sequence of eigensteps. Here, the key idea is to visualize eigenstep construction as iteratively building a staircase. This visualization leads to an algorithm, dubbed Top Kill, which produces a valid sequence of eigensteps whenever it is possible to do so. We then build on Top Kill to explicitly parametrize the set of all valid eigensteps. This yields an explicit method for constructing all self-adjoint matrices with a given spectrum and diagonal...

## Coherent States of Accelerated Relativistic Quantum Particles, Vacuum Radiation and the Spontaneous Breakdown of the Conformal SU(2,2) Symmetry

Calixto, M.; Perez-Romero, E.; Aldaya, V.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.4%
We give a quantum mechanical description of accelerated relativistic particles in the framework of Coherent States (CS) of the (3+1)-dimensional conformal group SU(2,2), with the role of accelerations played by special conformal transformations and with the role of (proper) time translations played by dilations. The accelerated ground state $\tilde\phi_0$ of first quantization is a CS of the conformal group. We compute the distribution function giving the occupation number of each energy level in $\tilde\phi_0$ and, with it, the partition function Z, mean energy E and entropy S, which resemble that of an "Einstein Solid". An effective temperature T can be assigned to this "accelerated ensemble" through the thermodynamic expression dE/dS, which leads to a (non linear) relation between acceleration and temperature different from Unruh's (linear) formula. Then we construct the corresponding conformal-SU(2,2)-invariant second quantized theory and its spontaneous breakdown when selecting Poincar\'e-invariant degenerated \theta-vacua (namely, coherent states of conformal zero modes). Special conformal transformations (accelerations) destabilize the Poincar\'e vacuum and make it to radiate.; Comment: 25 pages, LaTeX, 3 figures. Additional information (resulting in four extra pages) and a slight change of focus has been introduced in order to make the line of arguments more clear. Title changed accordingly

## Quadrature rules and distribution of points on manifolds

Brandolini, Luca; Choirat, Christine; Colzani, Leonardo; Gigante, Giacomo; Seri, Raffaello; Travaglini, Giancarlo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
We study the error in quadrature rules on a compact manifold. As in the Koksma-Hlawka inequality, we consider a discrepancy of the sampling points and a generalized variation of the function. In particular, we give sharp quantitative estimates for quadrature rules of functions in Sobolev classes.

## Implementation of discretized Gabor frames and their duals

Kloos, Tobias; Stöckler, Joachim; Gröchenig, Karlheinz
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
The usefulness of Gabor frames depends on the easy computability of a suitable dual window. This question is addressed under several aspects: several versions of Schulz's iterative algorithm for the approximation of the canonical dual window are analyzed for their numerical stability. For Gabor frames with totally positive windows or with exponential B-splines a direct algorithm yields a family of exact dual windows with compact support. It is shown that these dual windows converge exponentially fast to the canonical dual window.; Comment: 16 pages, 4 figures

## Surgery of spline-type and molecular frames

Romero, José Luis
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.4%
We prove a result about producing new frames for general spline-type spaces by piecing together portions of known frames. Using spline-type spaces as models for the range of certain integral transforms, we obtain results for time-frequency decompositions and sampling.; Comment: 34 pages. Corrected typos

## Geometry of the Welch Bounds

Datta, Somantika; Howard, Stephen; Cochran, Douglas
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.4%
A geometric perspective involving Grammian and frame operators is used to derive the entire family of Welch bounds. This perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k-tensors, tight frames, homogeneous polynomials, and t-designs. In particular. a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k-tensors. It is also shown that tightness of the bounds requires tight frames. The lack of tight frames in symmetric k-tensors in many cases, however, leads to consideration of sets that come as close as possible to attaining the bounds. The geometric derivation is then extended in the setting of generalized or continuous frames. The Welch bounds for finite sets and countably infinite sets become special cases of this general setting.; Comment: changes from previous version include: correction of typos, additional references added, new Example 3.3

## Asymptotics of orthogonal polynomials beyond the scope of Szego's theorem

Peherstorfer, F.; Volberg, A.; Yuditskii, P.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
First we give here a simple proof of a remarkable result of Videnskii and Shirokov: let $B$ be a Blaschke product with $n$ zeros, then there exists an outer function $\phi, \phi(0)=1$, such that $\|(B\phi)'\| \leq C n$, where $C$ is an absolute constant. Then we apply this result to a certain problem of finding the asymptotic of orthogonal polynomials.

## The Schur-Horn theorem for operators and frames with prescribed norms and frame operator

Antezana, J.; Massey, P.; Ruiz, M.; Stojanoff, D.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.4%
Let $\mathcal H$ be a Hilbert space. Given a bounded positive definite operator $S$ on $\mathcal H$, and a bounded sequence $\mathbf{c} = \{c_k \}_{k \in \mathbb N}$ of non negative real numbers, the pair $(S, \mathbf{c})$ is frame admissible, if there exists a frame $\{f_k \}_{k \in \mathbb{N}}$ on $\mathcal H$ with frame operator $S$, such that $\|f_k \|^2 = c_k$, $k \in \mathbb {N}$. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended version of Schur-Horn theorem, due to A. Neumann. We use it to get necessary conditions (and to generalize known sufficient conditions) for a pair $(S, \mathbf{c})$, to be frame admissible.; Comment: To appear in Illinois Journal of Math

## Orthogonal polynomials of discrete variable and boundedness of Dirichlet kernel

Obermaier, Josef; Szwarc, Ryszard
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.4%
For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal expansions with respect to $L^\infty$ norm, which generalize analogous results obtained for little $q$-Legendre, little $q$-Jacobi and little $q$-Laguerre polynomials, by the authors.

## Wigner measures in the discrete setting: high-frequency analysis of sampling & reconstruction operators

Macia, Fabricio
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.4%
The goal of this article is that of understanding how the oscillation and concentration effects developed by a sequence of functions in $\mathbb{R}^{d}$ are modified by the action of Sampling and Reconstruction operators on regular grids. Our analysis is performed in terms of Wigner and defect measures, which provide a quantitative description of the high frequency behavior of bounded sequences in $L^{2}(mathbb{R}^{d})$. We actually present explicit formulas that make possible to compute such measures for sampled/reconstructed sequences. As a consequence, we are able to characterize sampling and reconstruction operators that preserve or filter the high-frequency behavior of specific classes of sequences. The proofs of our results rely on the construction and manipulation of Wigner measures associated to sequences of discrete functions.

## Manifold structure of spaces of spherical tight frames

Dykema, Ken; Strawn, Nate
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.4%
We consider the space F^E_{k,n} of all spherical tight frames of k vectors in real or complex n--dimensional Hilbert space E^n, i.e. E=R or E=C, and its orbit space G^E_{k,n}=F^E_{k,n}/O^E_n under the obvious action of the group O^E_n of structure preserving transformations of E^n. We show that the quotient map F^E_{k,n} -> G^E_{k,n} is a locally trivial fiber bundle (also in the more general case of ellipsoidal tight frames) and that there is a homeomorphism G^E_{k,n} -> G^E_{k,k-n}. We show that G^E_{k,n} and F^E_{k,n} are real manifolds whenever k and n are relatively prime, and we describe them as disjoint unions of finitely many manifolds (of various dimensions) when when k and n have a common divisor. We also prove that F^R_{k,2} is connected (k >= 4) and F^R_{n+2,n} is connected, (n >= 2). The spaces G^R_{4,2} and G^R_{5,2} are investigated in detail. The former is found to be a graph and the latter is the orientable surface of genus 25.; Comment: The new version corrects some typographical errors, including a misleading error in the abstract: we show connectedness of F^R_{k,2}, not of more general F^R_{k,n}

## Projective multiresolution analyses for $L^2(R^2)$

Packer, Judith A.; Rieffel, Marc A.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.4%
We define the notion of "projective" multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra $C(\btn)$ of continuous complex-valued functions on an $n$-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyses, including the frames which they provide for $L^2(\brn)$. Then we show how to construct examples for the case of any diagonal $2 \times 2$ dilation matrix with integer entries, with initial module specified to be any fixed finitely generated projective $C(\mathbb T^2)$-module. We compute the isomorphism classes of the corresponding wavelet modules.; Comment: 25 pages

## Rank-One Decomposition of Operators and Construction of Frames

Kornelson, Keri; Larson, David