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## ‣ Paley-Wiener description of K-spherical Besov spaces on the Heisenberg group

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/11/2011
Português

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We characterize the Besov spaces associated to the Gelfand pairs on the
Heisenberg group. The characterization is given in terms of bandlimited wavelet
coefficients where the bandlimitedness is introduced using spherical Fourier
transform. To obtain these results we develop an approach to the
characterization of Besov spaces in abstract Hilbert spaces through compactly
supported admissible functions.; Comment: Keywords: Besov spaces, Paley-Wiener spaces, Gelfand pair, the
Heisenberg group, spherical transform, wavelets

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## ‣ Mexican Hat Wavelet on the Heisenberg Group

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/05/2007
Português

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In this article wavelets (admissible vectors) on the Heisenberg group are
studied from the point of view of Calderon's formula. Further we shall show
that for the class of Schwartz functions the Calderon admissibility condition
is equivalent to the usual admissibility property which will be introduced in
this work. Furthermore motivated by a well-known example on the real line, the
Mexican-Hat wavelet, we demonstrate the existence and construction of an
analogous wavelet on the Heisenberg Lie group with 2 vanishing moments, which
together with all of its derivatives has Gaussian decay.; Comment: 8 pages, no figures

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## ‣ A Haar-type Approximation and a New Numerical Schema for the Korteweg-de Vries Equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/07/2011
Português

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We discuss a new numerical schema for solving the initial value problem for
the Korteweg-de Vries equation for large times. Our approach is based upon the
Inverse Scattering Transform that reduces the problem to calculating the
reflection coefficient of the corresponding Schr\"odinger equation. Using a
step-like approximation of the initial profile and a fragmentation principle
for the scattering data, we obtain an explicit recursion formula for computing
the reflection coefficient, yielding a high resolution KdV solver. We also
discuss some generalizations of this algorithm and how it might be improved by
using Haar and other wavelets.

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## ‣ Certain representations of the Cuntz relations, and a question on wavelets decompositions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We compute the Coifman-Meyer-Wickerhauser measure $\mu$ for certain families
of quadrature mirror filters (QMFs), and we establish that for a subclass of
QMFs, $\mu$ contains a fractal scale. In particular, these measures $\mu$ are
not in the Lebesgue class.; Comment: v.2 has a new title and additional material in the introduction.
Prepared using the amsproc.cls document class

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## ‣ Gabor fields and wavelet sets for the Heisenberg group

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We study singly-generated wavelet systems on $\Bbb R^2$ that are naturally
associated with rank-one wavelet systems on the Heisenberg group $N$. We prove
a necessary condition on the generator in order that any such system be a
Parseval frame. Given a suitable subset $I$ of the dual of $N$, we give an
explicit construction for Parseval frame wavelets that are associated with $I$.
We say that $g\in L^2(I\times \Bbb R)$ is Gabor field over $I$ if, for a.e.
$\lambda \in I$, $|\lambda|^{1/2} g(\lambda,\cdot)$ is the Gabor generator of a
Parseval frame for $L^2(\Bbb R)$, and that $I$ is a Heisenberg wavelet set if
every Gabor field over $I$ is a Parseval frame (mother-)wavelet for $L^2(\Bbb
R^2)$. We then show that $I$ is a Heisenberg wavelet set if and only if $I$ is
both translation congruent with a subset of the unit interval and dilation
congruent with the Shannon set.

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## ‣ Interpolation maps and congruence domains for wavelet sets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/10/2007
Português

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It is proven that if an interpolation map between two wavelet sets preserves
the union of the sets, then the pair must be an interpolation pair. We also
construct an example of a pair of wavelet sets for which the congruence domains
of the associated interpolation map and its inverse are equal, and yet the pair
is not an interpolation pair. The first result solves affirmatively a problem
that the second author had posed several years ago, and the second result
solves an intriguing problem of D. Han. The key to this counterexample is a
special technical lemma on constructing wavelet sets. Several other
applications of this result are also given. In addition, some problems are
posed. We also take the opportunity to give some general exposition on wavelet
sets and operator-theoretic interpolation of wavelets.; Comment: 21 pages, 2 figures

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## ‣ A characterization of product BMO by commutators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Let b be a function on the plane. Let H_j, j=1,2, be the Hilbert transform
acting on the j-th coordinate on the plane. We show that the operator norm of
the double commutator [[ M_b, H_1], H_2] is equivalent to the Chang-Fefferman
BMO norm of b. Here, M_b denotes the operator which is multiplication by b.
This result extends a well known theorem of Nehari on weak factorization in
the Hardy space H^1 to the same theorem on H^1 of a product domain. The product
setting is more delicate because of the presence of a two parameter family of
dilations. The method of proof depends upon (a) A dyadic decomposition of
product BMO by wavelets (b) a prior estimate of Ferguson and Sadosky involving
rectangular BMO (c) and a careful control of certain measures related to those
of Carleson.; Comment: 21 pages. To appear in Acta Math. Appendix corrected

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## ‣ Low-pass filters and representations of the Baumslag Solitar group

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Classical Analysis and ODEs#Mathematics - Operator Algebras#42C40, 28A78, 46L45, 28D05, 22D25

We analyze representations of solvable Baumslag Solitar group that admit
wavelets and show how such representations can be constructed from a given
low-pass filter. We describe the direct integral decomposition for some
examples and derive from it a general criterion for the existence of solutions
for scaling equations. As another application, we construct a Fourier transform
for some Hausdorff measures.

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## ‣ $p$-Adic Haar multiresolution analysis and pseudo-differential operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/05/2007
Português

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#Mathematical Physics#Mathematics - General Mathematics#(Primary) 11F85, 42C40, 47G30#(Secondary) 26A33, 46F10

The notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We
discuss a ``natural'' refinement equation whose solution (a refinable function)
is the characteristic function of the unit disc. This equation reflects the
fact that the characteristic function of the unit disc is a sum of $p$
characteristic functions of mutually disjoint discs of radius $p^{-1}$. This
refinement equation generates a MRA. The case $p=2$ is studied in detail. Our
MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real
setting, the refinable function generating our Haar MRA is 1-periodic, which
never holds for real refinable functions. This fact implies that there exist
infinity many different 2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$
generated by the same Haar MRA. All of these bases are described. We also
constructed multidimensional 2-adic Haar orthonormal bases for
${\cL}^2(\bQ_2^n)$ by means of the tensor product of one-dimensional MRAs. A
criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a
pseudo-differential operator is derived. We proved also that these wavelets are
eigenfunctions of the Taibleson multidimensional fractional operator. These
facts create the necessary prerequisites for intensive using our bases in
applications.

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## ‣ Paraproducts and Products of functions in $BMO(\mathbb R^n)$ and $H^1(\mathbb R^n)$ through wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/03/2011
Português

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In this paper, we prove that the product (in the distribution sense) of two
functions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may be
written as the sum of two continuous bilinear operators, one from
$\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$, the other one from
$\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of Hardy-Orlicz space denoted
by $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the set
of distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies
$$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathcal
Mf(x)|)}dx <\infty.$$ The two bilinear operators can be defined in terms of
paraproducts. As a consequence, we find an endpoint estimate involving the
space $\H^{\log}(\bR^n)$ for the $\div$-$\curl$ lemma.

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## ‣ The near shift-invariance of the dual-tree complex wavelet transform revisited

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/04/2013
Português

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The dual-tree complex wavelet transform (DTCWT) is an enhancement of the
conventional discrete wavelet transform (DWT) due to a higher degree of
shift-invariance and a greater directional selectivity, finding its
applications in signal and image processing. This paper presents a quantitative
proof of the superiority of the DTCWT over the DWT in case of modulated
wavelets.; Comment: 15 pages

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## ‣ Reproducing subgroups of $Sp(2,\mathbb{R})$. Part I: algebraic classification

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We classify the connected Lie subgroups of the symplectic group
$Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form.
The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study
is motivated by the need of a unified approach to continuous 2D signal
analyses, as those provided by wavelets and shearlets.; Comment: 26 pages

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## ‣ p-Adic refinable functions and MRA-based wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/11/2007
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We described a wide class of $p$-adic refinable equations generating $p$-adic
multiresolution analysis. A method for the construction of $p$-adic orthogonal
wavelet bases within the framework of the MRA theory is suggested. A
realization of this method is illustrated by an example, which gives a new
3-adic wavelet basis. Another realization leads to the $p$-adic Haar bases
which were known before.; Comment: 10 pages

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## ‣ On alternative wavelet reconstruction formula: a case study of approximate wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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The application of the continuous wavelet transform to study of a wide class
of physical processes with oscillatory dynamics is restricted by large central
frequencies due to the admissibility condition. We propose an alternative
reconstruction formula for the continuous wavelet transform, which is
applicable even if the admissibility condition is violated. The case of the
transform with the standard Morlet wavelet, which is an important example of
such analyzing functions, is discussed.; Comment: 6 pages

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## ‣ A Discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmetric spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/04/2011
Português

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Let $f$ be a Paley-Wiener function in the space $L_{2}(X)$, where $X$ is a
symmetric space of noncompact type. It is shown that by using the values of $f$
on a sufficiently dense and separated set of points of $X$ one can give an
exact formula for the Helgason-Fourier transform of $f$. In order to find a
discrete approximation to the Helgason-Fourier transform of a function from a
Besov space on $X$ we develop an approximation theory by Paley-Wiener functions
in $L_{2}(X)$.; Comment: Radon transforms, geometry, and wavelets, 231-247, Contemp. Math.,
464, Amer. Math. Soc., Providence, RI, 2008

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## ‣ Almost diagonal matrices and Besov-type spaces based on wavelet expansions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/08/2014
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#Mathematics - Functional Analysis#Mathematics - Numerical Analysis#35B65, 42C40, 45E99, 46A45, 46E35, 47B37, 47B38, 65T60

This paper is concerned with problems in the context of the theoretical
foundation of adaptive (wavelet) algorithms for the numerical treatment of
operator equations. It is well-known that the analysis of such schemes
naturally leads to function spaces of Besov type. But, especially when dealing
with equations on non-smooth manifolds, the definition of these spaces is not
straightforward. Nevertheless, motivated by applications, recently Besov-type
spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$ on certain two-dimensional, patchwise
smooth surfaces were defined and employed successfully. In the present paper,
we extend this definition (based on wavelet expansions) to a quite general
class of $d$-dimensional manifolds and investigate some analytical properties
(such as, e.g., embeddings and best $n$-term approximation rates) of the
resulting quasi-Banach spaces. In particular, we prove that different prominent
constructions of biorthogonal wavelet systems $\Psi$ on domains or manifolds
$\Gamma$ which admit a decomposition into smooth patches actually generate the
same Besov-type function spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$, provided that
their univariate ingredients possess a sufficiently large order of cancellation
and regularity (compared to the smoothness parameter $\alpha$ of the space).
For this purpose...

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## ‣ Subsampling needlet coefficients on the sphere

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In a recent paper, we analyzed the properties of a new kind of spherical
wavelets (called needlets) for statistical inference procedures on spherical
random fields; the investigation was mainly motivated by applications to
cosmological data. In the present work, we exploit the asymptotic uncorrelation
of random needlet coefficients at fixed angular distances to construct
subsampling statistics evaluated on Voronoi cells on the sphere. We illustrate
how such statistics can be used for isotropy tests and for bootstrap estimation
of nuisance parameters, even when a single realization of the spherical random
field is observed. The asymptotic theory is developed in detail in the high
resolution sense.; Comment: Published in at http://dx.doi.org/10.3150/08-BEJ164 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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## ‣ Nonlinear Inversion from Partial EIT Data: Computational Experiments

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Numerical Analysis#Mathematics - Analysis of PDEs#Primary 65N21, 35R30, Secondary 45Q05

Electrical impedance tomography (EIT) is a non-invasive imaging method in
which an unknown physical body is probed with electric currents applied on the
boundary, and the internal conductivity distribution is recovered from the
measured boundary voltage data. The reconstruction task is a nonlinear and
ill-posed inverse problem, whose solution calls for special regularized
algorithms, such as D-bar methods which are based on complex geometrical optics
solutions (CGOs). In many applications of EIT, such as monitoring the heart and
lungs of unconscious intensive care patients or locating the focus of an
epileptic seizure, data acquisition on the entire boundary of the body is
impractical, restricting the boundary area available for EIT measurements. An
extension of the D-bar method to the case when data is collected only on a
subset of the boundary is studied by computational simulation. The approach is
based on solving a boundary integral equation for the traces of the CGOs using
localized basis functions (Haar wavelets). The numerical evidence suggests that
the D-bar method can be applied to partial-boundary data in dimension two and
that the traces of the partial data CGOs approximate the full data CGO
solutions on the available portion of the boundary...

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## ‣ Geometric Separation by Single-Pass Alternating Thresholding

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/04/2012
Português

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#Mathematics - Functional Analysis#Computer Science - Information Theory#Mathematics - Numerical Analysis

Modern data is customarily of multimodal nature, and analysis tasks typically
require separation into the single components. Although a highly ill-posed
problem, the morphological difference of these components sometimes allow a
very precise separation such as, for instance, in neurobiological imaging a
separation into spines (pointlike structures) and dendrites (curvilinear
structures). Recently, applied harmonic analysis introduced powerful
methodologies to achieve this task, exploiting specifically designed
representation systems in which the components are sparsely representable,
combined with either performing $\ell_1$ minimization or thresholding on the
combined dictionary.
In this paper we provide a thorough theoretical study of the separation of a
distributional model situation of point- and curvilinear singularities
exploiting a surprisingly simple single-pass alternating thresholding method
applied to the two complementary frames: wavelets and curvelets. Utilizing the
fact that the coefficients are clustered geometrically, thereby exhibiting
clustered/geometric sparsity in the chosen frames, we prove that at
sufficiently fine scales arbitrarily precise separation is possible. Even more
surprising, it turns out that the thresholding index sets converge to the
wavefront sets of the point- and curvilinear singularities in phase space and
that those wavefront sets are perfectly separated by the thresholding
procedure. Main ingredients of our analysis are the novel notion of cluster
coherence and clustered/geometric sparsity as well as a microlocal analysis
viewpoint.; Comment: 35 pages...

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## ‣ Optimal Decompositions of Translations of $L^{2}$-functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/11/2007
Português

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#Mathematics - Functional Analysis#Mathematics - Spectral Theory#47B40, 47B06, 06D22, 62M15, 42C40, 62M20

In this paper we offer a computational approach to the spectral function for
a finite family of commuting operators, and give applications. Motivated by
questions in wavelets and in signal processing, we study a problem about
spectral concentration of integral translations of functions in the Hilbert
space $L^{2}(\mathbb{R}^{n})$. Our approach applies more generally to families
of $n$ arbitrary commuting unitary operators in a complex Hilbert space
$\mathcal{H}$, or equivalent the spectral theory of a unitary representation
$U$ of the rank-$n$ lattice $\mathbb{Z}^{n}$ in $\mathbb{R}^{n}$. Starting with
a non-zero vector $\psi \in \mathcal{H}$, we look for relations among the
vectors in the cyclic subspace in $\mathcal{H}$ generated by $\psi$. Since
these vectors $\{U(k)\psi | k \in \mathbb{Z}^{n}\}$ involve infinite ``linear
combinations," the problem arises of giving geometric characterizations of
these non-trivial linear relations. A special case of the problem arose
initially in work of Kolmogorov under the name $L^{2}$-independence. This
refers to \textit{infinite} linear combinations of integral translates of a
fixed function with $l^{2}$-coefficients. While we were motivated by the study
of translation operators arising in wavelet and frame theory...

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