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## Sparsity Equivalence of Anisotropic Decompositions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/01/2011
Português

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#Mathematics - Functional Analysis#Computer Science - Numerical Analysis#Mathematics - Numerical Analysis#42C40 (Primary), 42C15, 65T60, 94A08 (Secondary)

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

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## Discretization of continuous frame

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/01/2011
Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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## Gabor fusion frames generated by difference sets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/07/2015
Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/09/2008
Português

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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 $X_0$, $\Theta_0$, $X_F$.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/12/2012
Português

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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.

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## Approximation of Fourier Integral Operators by Gabor multipliers

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/07/2011
Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/07/2011
Português

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#General Relativity and Quantum Cosmology#High Energy Physics - Theory#Mathematical Physics#81R30, 81R05, 42B05, 30H20, 42C15

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

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## Quadrature rules and distribution of points on manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/12/2010
Português

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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.

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## Implementation of discretized Gabor frames and their duals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/06/2015
Português

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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

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## Surgery of spline-type and molecular frames

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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## Geometry of the Welch Bounds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/08/2005
Português

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/07/2003
Português

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#Mathematics - Numerical Analysis#Mathematical Physics#Mathematics - Functional Analysis#42C15#94A12#65D05#46E35#46E39

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.

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## Manifold structure of spaces of spherical tight frames

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Functional Analysis#Mathematics - Algebraic Geometry#Mathematics - Operator Algebras#42C15#94A12#14P05

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}

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/09/2003
Português

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The construction of frames for a Hilbert space H can be equated to the
decomposition of the frame operator as a sum of positive operators having rank
one. This realization provides a different approach to questions regarding
frames with particular properties and motivates our results. We find a
necessary and sufficient condition under which any positive finite-rank
operator B can be expressed as a sum of rank-one operators with norms specified
by a sequence of positive numbers {c_i}. Equivalently, this result proves the
existence of a frame with B as it's frame operator and with vector norms given
by the square roots of the sequence elements. We further prove that, given a
non-compact positive operator B on an infinite dimensional separable real or
complex Hilbert space, and given an infinite sequence {c_i} of positive real
numbers which has infinite sum and which has supremum strictly less than the
essential norm of B, there is a sequence of rank-one positive operators, with
norms given by {c_i}, which sum to B in the strong operator topology.
These results generalize results by Casazza, Kovacevic, Leon, and Tremain, in
which the operator is a scalar multiple of the identity operator (or
equivalently the frame is a tight frame)...

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