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## ‣ Wavelets and PDEs : the improvement of computational performance using multi-resolution analysis; Wavelets and partial differential equations

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 125 p.; 8052978 bytes; 8052737 bytes; application/pdf; application/pdf

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by Dmitri Betaneli.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1998.; Includes bibliographical references (p. 119-125).

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## ‣ Recuperación de imágenes tomográficas con ruido aplicando Wavelets

Fonte: Universidad EAFIT; Maestría en Matemáticas Aplicadas; Escuela de Ciencias y Humanidades. Departamento de Ciencias Básicas
Publicador: Universidad EAFIT; Maestría en Matemáticas Aplicadas; Escuela de Ciencias y Humanidades. Departamento de Ciencias Básicas

Tipo: masterThesis; Tesis de Maestría; acceptedVersion

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#Transformada de Radon#Transformada de Fourier y Wavelet#Tomografía Computarizada#TRANSFORMACIONES DE FOURIER#TOMOGRAFÍA COMPUTARIZADA POR RAYOS X#FILTROS DIGITALES (MATEMÁTICAS)#ONDITAS (MATEMÁTICAS)#TEORÍA DE LAS DISTRIBUCIONES (ANÁLISIS FUNCIONAL)#ECUACIONES DE LA RECTA#ALGORITMOS#TOMOGRAFÍA

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## ‣ Making Pulsed-Beam Wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/01/2003
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#Mathematical Physics#Mathematics - Analysis of PDEs#Mathematics - Complex Variables#31-XX, 32-XX, 35-XX, 41-XX, 44-XX, 78-XX, 83-XX

Point sources in complex spacetime, which generate acoustic and
electromagnetic pulsed-beam wavelets, are rigorously defined and computed with
a view toward their realization.; Comment: 20 pages. Invited talk, "Multiscale Geometric Analysis" workshop, <
http://www.ipam.ucla.edu/programs/mga2003/>

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## ‣ Asymptotics and numerics of zeros of polynomials that are related to Daubechies wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/10/1996
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We give asymptotic approximations of the zeros of certain high degree
polynomials. The zeros can be used to compute the filter coefficients in the
dilation equations which define the compactly supported orthogonal Daubechies
wavelets. Computational schemes are presented to obtain the numerical values
of the zeros within high precision.

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## ‣ Wavelets in weighted norm spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/10/2014
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We give a complete characterization of the classes of weight functions for
which the Haar wavelet system for $m$-dilations, $m= 2,3,\ldots$ is an
unconditional basis in $L^p(\mathbb{R},w)$. Particulary it follows that higher
rank Haar wavelets are unconditional bases in the weighted norm spaces
$L^p(\mathbb{R},w)$, where $w(x) = |x|^{r}, r>p-1$. These weights can have very
strong zeros at the origin. Which shows that the class of weight functions for
which higher rank Haar wavelets are unconditional bases is much richer than it
was supposed. One of main purposes of our study is to show that weights with
strong zeros should be considered if somebody is studying basis properties of a
given wavelet system in a weighted norm space.

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## ‣ Generalized Huygens principle with pulsed-beam wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematical Physics#Mathematics - Analysis of PDEs#Physics - Optics#30-XX#32-XX#35-XX#41-XX#44-XX#46XX#65-XX#78-XX

Huygens' principle has a well-known problem with back-propagation due to the
spherical nature of the secondary wavelets. We solve this by analytically
continuing the surface of integration. If the surface is a sphere of radius
$R$, this is done by complexifying $R$ to $R+ia$. The resulting complex sphere
is shown to be a real bundle of disks with radius $a$ tangent to the sphere.
Huygens' "secondary source points" are thus replaced by disks, and his
spherical wavelets by well-focused pulsed beams propagating outward. This
solves the back-propagation problem. The extended Huygens principle is a
completeness relation for pulsed beams, giving a representation of a general
radiation field as a superposition of such beams. Furthermore, it naturally
yields a very efficient way to compute radiation fields because all pulsed
beams missing a given observer can be ignored. Increasing $a$ sharpens the
focus of the pulsed beams, which in turn raises the compression of the
representation.; Comment: 49 pages, 14 figures

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## ‣ An equivalence relation on wavelets in higher dimensions associated with matrix dilations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/09/2002
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We introduce an equivalence relation on the set of single wavelets of
L^2(R^n) associated with an arbitrary dilation matrix. The corresponding
equivalence classes are characterized in terms of the support of the Fourier
transform of wavelets and it is shown that each of these classes is non-empty.; Comment: 9 pages

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## ‣ Compactly supported wavelets and representations of the Cuntz relations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/12/1999
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#Mathematics - Functional Analysis#46L60, 47D25, 42A16, 43A65 (Primary) 46L45, 42A65, 41A15 (Secondary)

We study the harmonic analysis of the quadrature mirror filters coming from
multiresolution wavelet analysis of compactly supported wavelets. It is known
that those of these wavelets that come from third order polynomials are
parametrized by the circle, and we compute that the corresponding filters
generate irreducible mutually disjoint representations of of the Cuntz algebra
$ O_{2} $ except at two points on the circle. One of the two exceptional points
corresponds to the Haar wavelet and the other is the unique point on the circle
where the father function defines a tight frame which is not an orthonormal
basis. At these two points the representation decomposes into two and three
mutually disjoint irreducible representations, respectively, and the two
representations at the Haar point are each unitarily equivalent to one of the
three representations at the other singular point.; Comment: AMS-LaTeX; 29 pages, 6 figures incorporating 23 EPS diagrams and 1
LaTeX picture

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## ‣ Regression in random design and Bayesian warped wavelets estimators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/08/2009
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In this paper we deal with the regression problem in a random design setting.
We investigate asymptotic optimality under minimax point of view of various
Bayesian rules based on warped wavelets and show that they nearly attain
optimal minimax rates of convergence over the Besov smoothness class
considered. Warped wavelets have been introduced recently, they offer very good
computable and easy-to-implement properties while being well adapted to the
statistical problem at hand. We particularly put emphasis on Bayesian rules
leaning on small and large variance Gaussian priors and discuss their
simulation performances comparing them with a hard thresholding procedure.

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## ‣ Wavelets with the Translation Invariance Property of Order N

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/02/2000
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All wavelets can be associated to a multiresolution like structure, i.e. an
incr easing sequence of subspaces of L^2(R). We consider the interaction of a
wavel et and the translation operator in terms of which of the subspaces in
this multi resolution like structure are invariant under the translation
operator. This ac tion defines the notion of the translation invariance
property of order n. In this paper we show that wavelets of all levels of
translation invariance exist, first for the classic case of dilation by 2, and
then for arbitrary integral di lation factors.; Comment: 17 pages; AMS-Latex

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## ‣ Linear Phase Perfect Reconstruction Filters and Wavelets with Even Symmetry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/12/2011
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Perfect reconstruction filter banks can be used to generate a variety of
wavelet bases. Using IIR linear phase filters one can obtain symmetry
properties for the wavelet and scaling functions. In this paper we describe all
possible IIR linear phase filters generating symmetric wavelets with any
prescribed number of vanishing moments. In analogy with the well known FIR
case, we construct and study a new family of wavelets obtained by considering
maximal number of vanishing moments for each fixed order of the IIR filter.
Explicit expressions for the coefficients of numerator, denominator, zeroes,
and poles are presented.
This new parameterization allows one to design linear phase quadrature mirror
filters with many other properties of interest such as filters that have any
preassigned set of zeroes in the stopband or that satisfy an almost
interpolating property.
Using Beylkin's approach, it is indicated how to implement these IIR filters
not as recursive filters but as FIR filters.

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## ‣ Methods from multiscale theory and wavelets applied to non-linear dynamics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We show how fundamental ideas from signal processing, multiscale theory and
wavelets may be applied to non-linear dynamics.
The problems from dynamics include iterated function systems (IFS), dynamical
systems based on substitution such as the discrete systems built on rational
functions of one complex variable and the corresponding Julia sets, and state
spaces of subshifts in symbolic dynamics. Our paper serves to motivate and
survey our recent results in this general area. Hence we leave out some proofs,
but instead add a number of intuitive ideas which we hope will make the subject
more accessible to researchers in operator theory and systems theory.; Comment: survey. v2: We have polished the writing and corrected some of the
cross references and citations; and v2 has an acknowledgment paragraph added.
Moreover, the ms has been converted to Birkhauser/OT style files. v3: added
discussion of general themes of operator theory and how they complement the
particular structure arising from multiscale theory, p. 22; regularized
subsection numbering throughout; corrected some misspellings

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## ‣ Multiple Multidimensional Morse Wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/11/2005
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We define a set of operators that localise a radial image in radial space and
radial frequency simultaneously. We find the eigenfunctions of this operator
and thus define a non-separable orthogonal set of radial wavelet functions that
may be considered optimally concentrated over a region of radial space and
radial scale space, defined via a doublet of parameters. We give analytic forms
for their energy concentration over this region. We show how the radial
function localisation operator can be generalised to an operator, localising
any square integrable function in two dimensional Euclidean space. We show that
the latter operator, with an appropriate choice of localisation region,
approximately has the same eigenfunctions as the radial operator. Based on the
radial wavelets we define a set of quaternionic valued wavelet functions that
can extract local orientation for discontinuous signals and both orientation
and phase structure for oscillatory signals. The full set of quaternionic
wavelet functions are component wise orthogonal; hence their statistical
properties are tractable, and we give forms for the variability of the
estimates of the local phase and orientation, as well as the local energy of
the image. By averaging estimates across wavelets...

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## ‣ Unitary systems, wavelet sets, and operator-theoretic interpolation of wavelets and frames

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/04/2006
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A wavelet is a special case of a vector in a separable Hilbert space that
generates a basis under the action of a collection, or system, of unitary
operators. We will describe the operator-interpolation approach to wavelet
theory using the local commutant of a system. This is really an abstract
application of the theory of operator algebras to wavelet theory. The concrete
applications of this method include results obtained using specially
constructed families of wavelet sets. A frame is a sequence of vectors in a
Hilbert space which is a compression of a basis for a larger space. This is not
the usual definition in the frame literature, but it is easily equivalent to
the usual definition. Because of this compression relationship between frames
and bases, the unitary system approach to wavelets (and more generally:
wandering vectors) is perfectly adaptable to frame theory. The use of the local
commutant is along the same lines as in the wavelet theory. Finally, we discuss
constructions of frames with special properties using targeted decompositions
of positive operators, and related problems.; Comment: This is a semi-expository article based on a series of tutorial talks
given by the author as part of the "Workshop on Functional and Harmonic
Analyses of Wavelets and Frames" held Aug 4-7...

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## ‣ About construction of orthogonal wavelets with compact support and with scaling coefficient N

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/05/2007
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In this paper a simple method of construction of scaling function $\phi (x)$
and orthogonal wavelets with the compact support for any natural coefficient of
scaling $N\ge 2$ is given. Examples of construction of wavelets for
coefficients of scaling N=2 and N=3 are produced.; Comment: LaTeX2e, 15 pages

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## ‣ A Family of Wavelets and a new Orthogonal Multiresolution Analysis Based on the Nyquist Criterion

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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A generalisation of the Shannon complex wavelet is introduced, which is
related to raised cosine filters. This approach is used to derive a new family
of orthogonal complex wavelets based on the Nyquist criterion for Intersymbolic
Interference (ISI) elimination. An orthogonal Multiresolution Analysis (MRA) is
presented, showing that the roll-off parameter should be kept below 1/3. The
pass-band behaviour of the Wavelet Fourier spectrum is examined. The left and
right roll-off regions are asymmetric; nevertheless the Q-constant analysis
philosophy is maintained. Finally, a generalisation of the (square root) raised
cosine wavelets is proposed.; Comment: 8 pages, 14 figures

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## ‣ Bandlimited Wavelets on the Heisenberg Group

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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Let $\mathbb{H}$ be the three-dimensional Heisenberg group. We introduce a
structure on the Heisenberg group which consists of the biregular
representation of $\mathbb{H\times H}$ restricted to some discrete subset of
$\mathbb{H\times H}$ and a free group of automorphisms $H$ singly generated and
acting semi-simply on $\mathbb{H}$. Using well-known theorems borrowed from
Gabor theory, we are able to construct simple and computable bandlimited
discrete wavelets on the Heisenberg group. Moreover, we provide necessary and
sufficient conditions for the existence of these wavelets.

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## ‣ From full rank subdivision schemes to multichannel wavelets: A constructive approach

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/03/2013
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In this paper, we describe some recent results obtained in the context of
vector subdivision schemes which possess the so-called full rank property. Such
kind of schemes, in particular those which have an interpolatory nature, are
connected to matrix refinable functions generating orthogonal multiresolution
analyses for the space of vector-valued signals. Corresponding multichannel
(matrix) wavelets can be defined and their construction in terms of a very
efficient scheme is given. Some examples illustrate the nature of these matrix
scaling functions/wavelets.

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## ‣ Wavelets and Quantum Algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/03/2000
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Wavelets, known to be useful in non-linear multi-scale processes and in
multi-resolution analysis, are shown to have a q-deformed algebraic structure.
The translation and dilation operators of the theory associate with any scaling
equation a non-linear, two parameter algebra. This structure can be mapped onto
the quantum group $su_{q}(2)$ in one limit, and approaches a Fourier series
generating algebra, in another limit. A duality between any scaling function
and its corresponding non-linear algebra is obtained. Examples for the Haar and
B-wavelets are worked out in detail.; Comment: 27 pages Latex, 3 figure ps

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## ‣ Implementation of the Wavelet-Galerkin method for boundary value problems

Fonte: Rochester Instituto de Tecnologia
Publicador: Rochester Instituto de Tecnologia

Tipo: Tese de Doutorado

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#Mechanical engineering#QA371 .S34 1998#Boundary value problems#Wavelets (Mathematics)#Differential equations

The objective of this work is to develop a systematic method of
implementing the Wavelet-Galerkin method for approximating solutions of
differential equations. The beginning of this project included understanding what
a wavelet is, and then becoming familiar with some of the applications. The
Wavelet-Galerkin method, as applied in this paper, does not use a wavelet at all.
In actuality, it uses the wavelet's scaling function. The distinction between the
two will be given in the following sections of this paper.
The sections of this thesis will include defining wavelets and their scaling
functions. This will give the reader valued insight to wavelets and Discrete
Wavelet Transforms (DWT). Following this will be a section defining the
Galerkin method. The purpose of this section will be to give the reader an
understanding of how weighted residual methods work, in particular, the Galerkin
Method. Next will be a section on how Scaling functions will be implemented in
the Galerkin method, forming the Wavelet-Galerkin Method.
The focus of this investigation will deal with solutions to a basic
homogeneous differential equation. The solution of this basic equation will be
analyzed using three separate, distinct methods, and then the results will be
compared. These methods include the Wavelet-Galerkin Method...

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