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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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## ‣ Generalized low-pass filters and multiresolution analyses

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/12/2008
Português

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We study generalized filters that are associated to multiplicity functions
and homomorphisms of the dual of an abelian group. These notions are based on
the structure of generalized multiresolution analyses. We investigate when the
Ruelle operator corresponding to such a filter is a pure isometry, and then use
that characterization to study the problem of when a collection of closed
subspaces, which satisfies all the conditions of a GMRA except the trivial
intersection condition, must in fact have a trivial intersection. In this
context, we obtain a generalization of a theorem of Bownik and Rzeszotnik.; Comment: 20 pages including bibliography

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## ‣ Multidimensional Tauberian theorems for wavelet and non-wavelet transforms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Functional Analysis#Primary 40E05, 41A27, 42C40, 46F12. Secondary 26A12, 26A16, 26B35,
41A60, 41A65, 46F05, 46F10

We study several Tauberian properties of regularizing transforms of tempered
distributions with values in Banach spaces, that is, transforms of the form
$M^{\mathbf{f}}_{\phi}(x,y)=(\mathbf{f}\ast\phi_{y})(x)$, where the kernel
$\phi$ is a test function and $\phi_{y}(\cdot)=y^{-n}\phi(\cdot/y)$. If the
zeroth moment of $\phi$ vanishes, it is a wavelet type transform; otherwise, we
say it is a non-wavelet type transform.
The first aim of this work is to show that the scaling (weak) asymptotic
properties of distributions are \emph{completely} determined by boundary
asymptotics of the regularizing transform plus natural Tauberian hypotheses.
Our second goal is to characterize the spaces of Banach space-valued tempered
distributions in terms of the transform $M^{\mathbf{f}}_{\phi}(x,y)$. We
investigate conditions which ensure that a distribution that a priori takes
values in locally convex space actually takes values in a narrower Banach
space. Special attention is paid to find the \emph{optimal} class of kernels
$\phi$ for which these Tauberian results hold.
We give various applications of our Tauberian theory in the pointwise and
(micro-)local regularity analysis of Banach space-valued distributions, and
develop a number of techniques which are specially useful when applied to
scalar-valued functions and distributions. Among such applications...

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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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## ‣ Stability Criterion for Convolution-Dominated Infinite Matrices

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/07/2009
Português

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Let $\ell^p$ be the space of all $p$-summable sequences on $\mathbb{Z}$. An
infinite matrix is said to have $\ell^p$-stability if it is bounded and has
bounded inverse on $\ell^p$. In this paper, a practical criterion is
established for the $\ell^p$-stability of convolution-dominated infinite
matrices.

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## ‣ Decompositions of Trigonometric Polynomials with Applications to Multivariate Subdivision Schemes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/07/2009
Português

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We study multivariate trigonometric polynomials, satisfying a set of
constraints close to the known Strung-Fix conditions. Based on the polyphase
representation of these polynomials relative to a general dilation matrix, we
develop a simple constructive method for a special type of decomposition of
such polynomials. These decompositions are of interest to the analysis of
convergence and smoothness of multivariate subdivision schemes associated with
general dilation matrices. We apply these decompositions, by verifying
sufficient conditions for the convergence and smoothness of multivariate scalar
subdivision schemes, proved here. For the convergence analysis our sufficient
conditions apply to arbitrary dilation matrices, while the previously known
necessary and sufficient conditions are relevant only in case of dilation
matrices with a self similar tiling. For the analysis of smoothness, we state
and prove two theorems on multivariate matrix subdivision schemes, which lead
to sufficient conditions for C^1 limits of scalar multivariate subdivision
schemes associated with isotropic dilation matrices. Although similar results
are stated in the literature, we give here detailed proofs of the results,
which we could not find elsewhere.; Comment: 30 pages

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## ‣ A duality approach to representations of Baumslag-Solitar groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Operator Algebras#Mathematics - Functional Analysis#22D20, 22D30, 37A15, 37A55, 42C40, 43A65, 46G15, 47D07

We give an operator theoretic approach to the constructions of
multiresolutions as they are used in a number of basis constructions with
wavelets, and in Hilbert spaces on fractals. Our approach starts with the
following version of the classical Baumslag-Solitar relations $u t = t^2 u$
where $t$ is a unitary operator in a Hilbert space $\mathcal H$ and $u$ is an
isometry in $\mathcal H$. There are isometric dilations of this system into a
bigger Hilbert space, relevant for wavelets. For a variety of carefully
selected dilations, the ``bigger'' Hilbert space may be $L^2(\br)$, and the
dilated operators may be the unitary operators which define a dyadic wavelet
multiresolutions of $L^2(\br)$ with the subspace $\mathcal H$ serving as the
corresponding resolution subspace. That is, the initialized resolution which is
generated by the wavelet scaling function(s). In the dilated Hilbert space, the
Baumslag-Solitar relations then take the more familiar form $u t u^{-1} = t^2$.
We offer an operator theoretic framework including the standard construction;
and we show how the representations of certain discrete semidirect group
products serve to classify the possibilities. For this we analyze and compare
several types of unitary representations of these semidirect products: the
induced representations in Mackey's theory...

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## ‣ Multivariate Wavelet Frames

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/04/2007
Português

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We proved that for any matrix dilation and for any positive integer $n$,
there exists a compactly supported tight wavelet frame with approximation order
$n$. Explicit methods for construction of dual and tight wavelet frames with a
given number of vanishing moments are suggested.; Comment: LaTex file, 28 pages

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## ‣ Shear Anisotropic Inhomogeneous Besov And Triebel-Lizorkin Spaces In $R^d$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/11/2012
Português

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We define distribution spaces of a sequence of convolutions of a set of
distributions with smooth functions, the shearlet system. Then, we define
associated sequence spaces and prove characterizations. We also show a
reproducing identity in the class of distributions. Finally, we prove
Sobolev-type embeddings within the shear anisotropic inhomogeneous spaces and
embeddings between (classical dyadic) isotropic inhomogeneous spaces and shear
anisotropic inhomogeneous spaces.; Comment: 36 pages. arXiv admin note: substantial text overlap with
arXiv:1203.5136

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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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## ‣ Convergence rate of wavelet expansions of Gaussian random processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/08/2013
Português

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The paper characterizes uniform convergence rate for general classes of
wavelet expansions of stationary Gaussian random processes. The convergence in
probability is considered.; Comment: This is an Author's Accepted Manuscript of an article to be published
in the Communications in Statistics - Theory and Methods. 24 pages. arXiv
admin note: substantial text overlap with arXiv:1307.2428

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## ‣ Uniform convergence of compactly supported wavelet expansions of Gaussian random processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/08/2013
Português

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New results on uniform convergence in probability for expansions of Gaussian
random processes using compactly supported wavelets are given. The main result
is valid for general classes of nonstationary processes. An application of the
obtained results to stationary processes is also presented. It is shown that
the convergence rate of the expansions is exponential.; Comment: This is an Author's Accepted Manuscript of an article published in
the Communications in Statistics - Theory and Methods. 15 pages

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## ‣ Bernuau spline wavelets and Sturmian sequences

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/09/2003
Português

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A spline wavelets construction of class C^n(R) supported by sequences of
aperiodic discretizations of R is presented. The construction is based on
multiresolution analysis recently elaborated by G. Bernuau. At a given scale,
we consider discretizations that are sets of left-hand ends of tiles in a
self-similar tiling of the real line with finite local complexity.
Corresponding tilings are determined by two-letter Sturmian substitution
sequences. We illustrate the construction with examples having quadratic
Pisot-Vijayaraghavan units (like tau = (1 + sqrt{5})/2 or tau^2 = (3 +
sqrt{5})/2) as scaling factor. In particular, we present a comprehensive
analysis of the Fibonacci chain and give the analytic form of related scaling
functions and wavelets as splines of second order. We also give some hints for
the construction of multidimensional spline wavelets based on stone-inflation
tilings in arbitrary dimension.; Comment: 41 pages,10 figures

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## ‣ Random Wavelet Series: Theory and Applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/10/2003
Português

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Random Wavelet Series form a class of random processes with multifractal
properties. We give three applications of this construction. First, we
synthesize a random function having any given spectrum of singularities
satisfying some conditions (but including non-concave spectra). Second, these
processes provide examples where the multifractal spectrum coincides with the
spectrum of large deviations, and we show how to recover it numerically.
Finally, particular cases of these processes satisfy a generalized
selfsimilarity relation proposed in the theory of fully developed turbulence.; Comment: To appear in Annales Math\'ematiques Blaise Pascal

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## ‣ Sampling theorems on locally compact groups from oscillation estimates

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/09/2005
Português

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#Mathematics - Functional Analysis#Mathematics - Representation Theory#Primary 43A80, 42B35#Secondary 42C40, 26D10

We present a general approach to derive sampling theorems on locally compact
groups from oscillation estimates. We focus on the ${\rm L}^2$-stability of the
sampling operator by using notions from frame theory. This approach yields
particularly simple and transparent reconstruction procedures. We then apply
these methods to the discretization of discrete series representations and to
Paley-Wiener spaces on stratified Lie groups.; Comment: 18 pages

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## ‣ Density of the Set of Generators of Wavelet Systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Given a function $\psi$ in $ \LL^2(\R^d)$, the affine (wavelet) system
generated by $\psi$, associated to an invertible matrix $a$ and a lattice
$\zG$, is the collection of functions $\{|\det a|^{j/2} \psi(a^jx-\gamma): j
\in \Z, \gamma \in \zG\}$. In this article we prove that the set of functions
generating affine systems that are a Riesz basis of $ \LL^2(\R^d)$ is dense in
$ \LL^2(\R^d)$.
We also prove that a stronger result is true for affine systems that are a
frame of $ \LL^2(\R^d)$. In this case we show that the generators associated to
a fixed but arbitrary dilation are a dense set.
Furthermore, we analyze the orthogonal case in which we prove that the set of
generators of orthogonal (not necessarily complete) affine systems, that are
compactly supported in frequency, are dense in the unit sphere of $
\LL^2(\R^d)$ with the induced metric. As a byproduct we introduce the
$p$-Grammian of a function and prove a convergence result of this Grammian as a
function of the lattice. This result gives insight in the problem of
oversampling of affine systems.; Comment: 15 pages-to appear in Constructive Approximation. Revised version

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## ‣ Use of operator algebras in the analysis of measures from wavelets and iterated function systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/09/2005
Português

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In this paper, we show how a class of operators used in the analysis of
measures from wavelets and iterated function systems may be understood from a
special family of representations of Cuntz algebras.

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## ‣ Groups, Wavelets, and Wavelet Sets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/07/2003
Português

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Wavelet and frames have become a widely used tool in mathematics, physics,
and applied science during the last decade. This article gives an overview over
some well known results about the continuous and discrete wavelet transforms
and groups acting on $\mathbb{R}^n$. We also show how this action can give rise
to wavelets, and in particular, MSF wavelets)in $L^2(\mathbb{R}^n)$.

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## ‣ Multiresolution in the Bergman space

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Complex Variables#Mathematics - Numerical Analysis#43A32, 42C40, 42C40, 33C47, 43A65, 41A20

In this paper we give a multiresolution construction in Bergman space. The
successful application of rational orthogonal bases needs a priori knowledge of
the poles of the transfer function that may cause a drawback of the method. We
give a set of poles and using them we will generate a multiresolution in $A^2$.
We study the upper and lower density of this set, and we give sufficient
conditions for this set to be interpolating or sampling sequence for the
Bergman space. The construction is an analogy with the discrete affine
wavelets, and in fact is the discretization of the continuous voice transform
generated by a representation of the Blaschke group over the Bergman space. The
constructed discretization scheme gives opportunity of practical realization of
hyperbolic wavelet representation of signals belonging to the Bergman space if
we can measure their values on a given set of points inside the unit disc.
Convergence properties of the hyperbolic wavelet representation will be
studied.; Comment: 15 pages, revised version

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## ‣ Adaptive Nonparametric Regression on Spin Fiber Bundles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/09/2010
Português

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#Mathematics - Statistics Theory#Astrophysics - Cosmology and Nongalactic Astrophysics#62G08, 62G20, 42B35, 42C10, 42C40, 46E35 42C40, 46E35

The construction of adaptive nonparametric procedures by means of wavelet
thresholding techniques is now a classical topic in modern mathematical
statistics. In this paper, we extend this framework to the analysis of
nonparametric regression on sections of spin fiber bundles defined on the
sphere. This can be viewed as a regression problem where the function to be
estimated takes as its values algebraic curves (for instance, ellipses) rather
than scalars, as usual. The problem is motivated by many important
astrophysical applications, concerning for instance the analysis of the weak
gravitational lensing effect, i.e. the distortion effect of gravity on the
images of distant galaxies. We propose a thresholding procedure based upon the
(mixed) spin needlets construction recently advocated by Geller and Marinucci
(2008,2010) and Geller et al. (2008,2009), and we investigate their rates of
convergence and their adaptive properties over spin Besov balls.; Comment: 40 pages

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