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## ‣ Avaliação de Técnicas de Interpolação de Imagens Digitais

Fonte: Universidade Estadual Paulista (UNESP)
Publicador: Universidade Estadual Paulista (UNESP)

Tipo: Dissertação de Mestrado
Formato: 138 f. : il.

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#Computação - Matematica#Interpolação#Formas quadraticas#Imagens digitais#Wavelets (Matematica)#Computer science - Mathematics

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES); Pós-graduação em Matematica Aplicada e Computacional - FCT; Nesta dissertação é realizado um estudo comparativo sobre alguns tipos de algoritmos aplicados a imagens digitais voltado para interpolação. Este trabalho inclui os métodos clássicos, que são: replicação, bilinear, bicúbica, Lagrange e interpolação pela função sinc; e alguns recentes: algoritmo-localmente adaptativo, método New Edge-Direction Interpolation (NEDI), improved New Edge Direction Interpolation (iNEDI), iterative curvaturebased interpolation (ICBI), interpolação utilizando wavelets redundantes e utilizando filtro bilateral. Todos os novos métodos possuem melhorias em aspectos visuais e redução de ruídos nas bordas em relação aos clássicos. Os métodos avaliados são comparados visualmente e quantitativamente utilizando as métricas estatísticas: erro médio quadrático (MSE), Raíz do Erro Médio Quadrático (RMSE), Erro Médio Quadrático Normalizado (NMSE), Relação Sinal-Ruído (SNR), Coeficiente de Correlação (CC) e Índice de Qualidade Universal (IQI). Também é realizada uma discussão dos resultados obtidos, analisando as qualidades e os defeitos dos métodos estudados. Por fim...

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## ‣ Método Wavelet-Petrov-Galerkin en la solución numérica de la ecuación KdV

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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#Trabajo intelectual. Universidad EAFIT#Tesis. Maestría en Matemáticas Aplicadas#Transformada Wavelet#Transformada de Fourier#Analysis#General aspects of analysis#Sequences and series#Fourier and harmonic analysis#TRANSFORMACIONES DE FOURIER#ANALISIS DE FOURIER#SERIES DE FOURIER

Sin lugar a dudas los métodos wavelets permiten desarrollar algoritmos eficientes y novedosos en el estudio del procesamiento de imágenes y señales. La idea de utilizar esta teoría en la solución numérica de ecuaciones en derivadas parciales se da en virtud a que algunas propiedades de las wavelets son importantes en la construcción de algoritmos adaptativos. Un algoritmo de este tipo selecciona un conjunto minimal de aproximaciones en cada paso, de tal manera que la solución calculada sea lo suficientemente próxima a la solución exacta. Si queremos que la solución calculada sea suave en alguna región, sólo unos pocos coeficientes wavelet serán necesarios para obtener una buena aproximación de la solución en dicha región, es decir, solamente los coeficientes de bajas frecuencias cuyo soporte esté en esa región son los utilizados. De otro lado, los coeficientes grandes (en valor absoluto) se localizan cerca de las singularidades y esto nos permite definir criterios de adaptabilidad a través del tiempo de evaluación [15, 23, 53, 64]. Este trabajo se dirige fundamentalmente a encontrar soluciones aproximadas a problemas del tipo hiperbólico o parabólicos, utilizando el método wavelet-Galerkin. El trabajo busca dar respuesta problemas que surgen en diferentes áreas de las ciencias e ingeniería.; v...

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## ‣ Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/12/1996
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#Mathematics - Functional Analysis#Mathematics - Operator Algebras#46L55, 47C15 (Primary) 42C05, 22D25, 11B85 (Secondary)

In this paper we show how wavelets originating from multiresolution analysis
of scale N give rise to certain representations of the Cuntz algebras O_N, and
conversely how the wavelets can be recovered from these representations. The
representations are given on the Hilbert space L^2(T) by
(S_i\xi)(z)=m_i(z)\xi(z^N). We characterize the Wold decomposition of such
operators. If the operators come from wavelets they are shifts, and this can be
used to realize the representation on a certain Hardy space over L^2(T). This
is used to compare the usual scale-2 theory of wavelets with the scale-N
theory. Also some other representations of O_N of the above form called
diagonal representations are characterized and classified up to unitary
equivalence by a homological invariant.; Comment: 59 pages, AMS-LaTeX v1.2b

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## ‣ Smooth Parseval frames for $L^2(\mathbb{R})$ and generalizations to $L^2(\mathbb{R}^d)$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/10/2012
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Wavelet set wavelets were the first examples of wavelets that may not have
associated multiresolution analyses. Furthermore, they provided examples of
complete orthonormal wavelet systems in $L^2(\mathbb{R}^d)$ which only require
a single generating wavelet. Although work had been done to smooth these
wavelets, which are by definition discontinuous on the frequency domain,
nothing had been explicitly done over $\mathbb{R}^d$, $d >1$. This paper, along
with another one cowritten by the author, finally addresses this issue.
Smoothing does not work as expected in higher dimensions. For example, Bin
Han's proof of existence of Schwartz class functions which are Parseval frame
wavelets and approximate Parseval frame wavelet set wavelets does not easily
generalize to higher dimensions. However, a construction of wavelet sets in
$\hat{\mathbb{R}}^d$ which may be smoothed is presented. Finally, it is shown
that a commonly used class of functions cannot be the result of convolutional
smoothing of a wavelet set wavelet.; Comment: 15 pages

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## ‣ Covariant Transform

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Functional Analysis#Mathematics - Complex Variables#Mathematics - Representation Theory#43A85, 32M99, 43A32, 46E10, 47A60, 47A67, 47C99, 81R30

The paper develops theory of covariant transform, which is inspired by the
wavelet construction. It was observed that many interesting types of wavelets
(or coherent states) arise from group representations which are not square
integrable or vacuum vectors which are not admissible. Covariant transform
extends an applicability of the popular wavelets construction to classic
examples like the Hardy space H_2, Banach spaces, covariant functional calculus
and many others.
Keywords: Wavelets, coherent states, group representations, Hardy space,
Littlewood-Paley operator, functional calculus, Berezin calculus, Radon
transform, Moebius map, maximal function, affine group, special linear group,
numerical range, characteristic function, functional model.; Comment: 9 pages, LaTeX2e (AMS-LaTeX); v2: minor corrections

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## ‣ Wavelet transforms versus Fourier transforms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/03/1993
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This note is a very basic introduction to wavelets. It starts with an
orthogonal basis of piecewise constant functions, constructed by dilation and
translation. The ``wavelet transform'' maps each $f(x)$ to its coefficients
with respect to this basis. The mathematics is simple and the transform is
fast (faster than the Fast Fourier Transform, which we briefly explain), but
approximation by piecewise constants is poor. To improve this first wavelet,
we are led to dilation equations and their unusual solutions. Higher-order
wavelets are constructed, and it is surprisingly quick to compute with them
--- always indirectly and recursively. We comment informally on the contest
between these transforms in signal processing, especially for video and image
compression (including high-definition television). So far the Fourier
Transform --- or its 8 by 8 windowed version, the Discrete Cosine Transform
--- is often chosen. But wavelets are already competitive, and they are ahead
for fingerprints. We present a sample of this developing theory.; Comment: 18 pages

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## ‣ Operator-Like Wavelet Bases of $L_2(\mathbb{R}^d)$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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The connection between derivative operators and wavelets is well known. Here
we generalize the concept by constructing multiresolution approximations and
wavelet basis functions that act like Fourier multiplier operators. This
construction follows from a stochastic model: signals are tempered
distributions such that the application of a whitening (differential) operator
results in a realization of a sparse white noise. Using wavelets constructed
from these operators, the sparsity of the white noise can be inherited by the
wavelet coefficients. In this paper, we specify such wavelets in full
generality and determine their properties in terms of the underlying operator.; Comment: 34 pages

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## ‣ Relationships among Interpolation Bases of Wavelet Spaces and Approximation Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/12/2012
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A multiresolution analysis is a nested chain of related approximation
spaces.This nesting in turn implies relationships among interpolation bases in
the approximation spaces and their derived wavelet spaces. Using these
relationships, a necessary and sufficient condition is given for existence of
interpolation wavelets, via analysis of the corresponding scaling functions. It
is also shown that any interpolation function for an approximation space plays
the role of a special type of scaling function (an interpolation scaling
function) when the corresponding family of approximation spaces forms a
multiresolution analysis. Based on these interpolation scaling functions, a new
algorithm is proposed for constructing corresponding interpolation wavelets
(when they exist in a multiresolution analysis). In simulations, our theorems
are tested for several typical wavelet spaces, demonstrating our theorems for
existence of interpolation wavelets and for constructing them in a general
multiresolution analysis.

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## ‣ A Note on the Daubechies Approach in the Construction of Spline Type Orthogonal Scaling Functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/07/2015
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We use Lorentz polynomials to present the solutions explicitly of equations
(6.1.7) of [I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional
Conference Series in Applied Mathematics, 61. Society for Industrial and
Applied Mathematics (SIAM), Philadelphia, PA, 1992] and (4.9) of [I.
Daubechies, Orthonormal bases of compactly supported wavelets. Comm. Pure Appl.
Math. 41 (1988), no. 7, 909--996] sot that we give an efficient way to prove
Daubechies' results on the existence of spline type orthogonal scaling
functions and to evaluate Daubechies scaling functions.

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## ‣ Physical wavelets: Lorentz covariant, singularity-free, finite energy, zero action, localized solutions to the wave equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/04/2003
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Particle physics has for some time made extensive use of extended field
configuations such as solitons, instantons, and sphalerons. However, no direct
use has yet been made of the quite extensive literature on ``localized wave''
configurations developed by the engineering, optics, and mathematics
communities. In this article I will exhibit a particularly simple ``physical
wavelet'' -- it is a Lorentz covariant classical field configuration that lives
in physical Minkowski space. The field is everwhere finite and nonsingular, and
has quadratic falloff in both space and time. The total energy is finite, the
total action is zero, and the field configuration solves the wave equation.
These physical wavelets can be constructed for both complex and real scalar
fields, and can be extended to the Maxwell and Yang-Mills fields in a
straightforward manner. Since these wavelets are finite energy, they are
guaranteed to be classically present at finite temperature; since they are zero
action, they can contribute to the quantum mechanical path integral at zero
``cost''.; Comment: 12 pages, JHEP3.cls

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## ‣ Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Functional Analysis#Computer Science - Information Theory#Mathematics - Numerical Analysis#41A65, 68P30, 68U10

Recently, compressed sensing techniques in combination with both wavelet and
directional representation systems have been very effectively applied to the
problem of image inpainting. However, a mathematical analysis of these
techniques which reveals the underlying geometrical content is completely
missing. In this paper, we provide the first comprehensive analysis in the
continuum domain utilizing the novel concept of clustered sparsity, which
besides leading to asymptotic error bounds also makes the superior behavior of
directional representation systems over wavelets precise. First, we propose an
abstract model for problems of data recovery and derive error bounds for two
different recovery schemes, namely l_1 minimization and thresholding. Second,
we set up a particular microlocal model for an image governed by edges inspired
by seismic data as well as a particular mask to model the missing data, namely
a linear singularity masked by a horizontal strip. Applying the abstract
estimate in the case of wavelets and of shearlets we prove that -- provided the
size of the missing part is asymptotically to the size of the analyzing
functions -- asymptotically precise inpainting can be obtained for this model.
Finally, we show that shearlets can fill strictly larger gaps than wavelets in
this model.; Comment: 49 pages...

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## ‣ Close Approximations for Daublets and their Spectra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/02/2015
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This paper offers a new regard on compactly supported wavelets derived from
FIR filters. Although being continuous wavelets, analytical formulation are
lacking for such wavelets. Close approximations for daublets (Daubechies
wavelets) and their spectra are introduced here. The frequency detection
properties of daublets are investigated through scalograms derived from these
new analytical expressions. These near-daublets have been implemented on the
Matlab wavelet toolbox and a few scalograms presented. This approach can be
valuable for wavelet synthesis from hardware or for application involving
continuous wavelet-based systems, such as wavelet OFDM.; Comment: 6 pages, 6 figures, 3 tables. Conference: International
Telecommunication Symposium, ITS 2010, Manaus, AM , Brazil

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## ‣ Real-time dynamics acquisition from irregular samples -- with application to anesthesia evaluation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/06/2014
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The first objective of this paper is to introduce a unified approach to the
D/A conversion, a real-time algorithm referred to as {\it blending operator},
based on spline functions of arbitrarily desired order, to interpolate the
irregular data samples, while preserving all polynomials of the same spline
order, with assured maximum order of approximation. This helps remove the two
main obstacles for adapting the recently proposed time-frequency analysis
technique {\it Synchrosqueezing transform} (SST) to irregular data samples in
order to allow online computation. Secondly, for real-time dynamic information
extraction from an oscillatory signal via SST, a family of vanishing-moment and
minimum-supported spline-wavelets (to be called VM wavelets) are introduced for
on-line computation of the CWT and its derivative. The second objective of this
paper is to apply the proposed real-time algorithm and VM wavelets to clinical
applications, particularly to the study of the "anesthetic depth" of a patient
during surgery, with emphasis on analyzing two dynamic quantities: the
"instantaneous frequencies" and the "non-rhythmic to rhythmic ratios" of the
patient's respiration, based on a one-lead electrocardiogram (ECG) signal.It is
envisioned that the proposed algorithm and VM wavelets should enable real-time
monitoring of "anesthetic depth"...

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## ‣ Eigenwavelets of the Wave equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We study a class of localized solutions of the wave equation, called
eigenwavelets, obtained by extending its fundamental solutions to complex
spacetime in the sense of hyperfunctions. The imaginary spacetime variables y,
which form a timelike vector, act as scale parameters generalizing the scale
variable of wavelets in one dimension. They determine the shape of the wavelets
in spacetime, making them pulsed beams that can be focused as tightly as
desired around a single ray by letting y approach the light cone. Furthermore,
the absence of any sidelobes makes them especially attractive for
communications, remote sensing and other applications using acoustic waves. (A
similar set of "electromagnetic eigenwavelets" exists for Maxwell's equations.)
I review the basic ideas in Minkowski space, then compute sources whose
realization should make it possible to radiate and absorb such wavelets. This
motivates an extension of Huygens' principle allowing equivalent sources to be
represented on shells instead of surfaces surrounding a bounded source.; Comment: 15 pages, 4 figures, invited paper, conference honoring Carlos
Berenstein

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## ‣ Wavelet transform and Radon transform on the Quaternion Heisenberg group

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/10/2011
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Let $\mathscr Q$ be the quaternion Heisenberg group, and let $\mathbf P$ be
the affine automorphism group of $\mathscr Q$. We develop the theory of
continuous wavelet transform on the quaternion Heisenberg group via the unitary
representations of $\mathbf P$ on $L^2(\mathscr Q)$. A class of radial wavelets
is constructed. The inverse wavelet transform is simplified by using radial
wavelets. Then we investigate the Radon transform on $\mathscr Q$. A
Semyanistri-Lizorkin space is introduced, on which the Radon transform is a
bijection. We deal with the Radon transform on $\mathscr Q$ both by the
Euclidean Fourier transform and the group Fourier transform. These two
treatments are essentially equivalent. We also give an inversion formula by
using wavelets, which does not require the smoothness of functions if the
wavelet is smooth.

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## ‣ Shannon wavelet approximations of linear differential operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/01/2007
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Recent works emphasized the interest of numerical solution of PDE's with
wavelets. In their works, A.Cohen, W.Dahmen and R.DeVore focussed on the non
linear approximation aspect of the wavelet approximation of PDE's to prove the
relevance of such methods. In order to extend these results, we focuss on the
convergence of the iterative algorithm, and we consider different possibilities
offered by the wavelet theory: the tensorial wavelets and the
derivation/integration of wavelet bases. We also investigate the use of wavelet
packets. We apply these extended results to prove in the case of the Shannon
wavelets, the convergence of the Leray projector algorithm with divergence-free
wavelets.; Comment: preprint IMPAN (19 pages)

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## ‣ Deformations of Gabor Frames

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/08/2001
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The quantum mechanical harmonic oscillator Hamiltonian generates a
one-parameter unitary group W(\theta) in L^2(R) which rotates the
time-frequency plane. In particular, W(\pi/2) is the Fourier transform. When
W(\theta) is applied to any frame of Gabor wavelets, the result is another such
frame with identical frame bounds. Thus each Gabor frame gives rise to a
one-parameter family of frames, which we call a deformation of the original.
For example, beginning with the usual tight frame F of Gabor wavelets generated
by a compactly supported window g(t) and parameterized by a regular lattice in
the time-frequency plane, one obtains a family of frames F_\theta generated by
the non-compactly supported windows g_\theta=W(theta)g, parameterized by
rotated versions of the original lattice. This gives a method for constructing
tight frames of Gabor wavelets for which neither the window nor its Fourier
transform have compact support. When \theta=\pi/2, we obtain the well-known
Gabor frame generated by a window with compactly supported Fourier transform.
The family F_\theta therefore interpolates these two familiar examples.; Comment: 8 pages in Plain Tex

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## ‣ The Hyperanalytic Wavelet Transform

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/05/2006
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In this paper novel classes of 2-D vector-valued spatial domain wavelets are
defined, and their properties given. The wavelets are 2-D generalizations of
1-D analytic wavelets, developed from the Generalized Cauchy-Riemann equations
and represented as quaternionic functions. Higher dimensionality complicates
the issue of analyticity, more than one `analytic' extension of a real function
is possible, and an `analytic' analysis wavelet will not necessarily construct
`analytic' decomposition coefficients. The decomposition of locally
unidirectional and/or separable variation is investigated in detail, and two
distinct families of hyperanalytic wavelet coefficients are introduced, the
monogenic and the hypercomplex wavelet coefficients. The recasting of the
analysis in a different frame of reference and its effect on the constructed
coefficients is investigated, important issues for sampled transform
coefficients. The magnitudes of the coefficients are shown to exhibit stability
with respect to shifts in phase. Hyperanalytic 2-D wavelet coefficients enable
the retrieval of a phase-and-magnitude description of an image in phase space,
similarly to the description of a 1-D signal with the use of 1-D analytic
wavelets, especially appropriate for oscillatory signals. Existing 2-D
directional wavelet decompositions are related to the newly developed
framework...

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## ‣ Wavelet Characterizations of the Atomic Hardy Space $H^1$ on Spaces of Homogeneous Type

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/09/2015
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#Mathematics - Classical Analysis and ODEs#Mathematics - Functional Analysis#Primary 42B30, Secondary 42C40, 30L99

Let $({\mathcal X},d,\mu)$ be a metric measure space of homogeneous type in
the sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be the
atomic Hardy space. Via orthonormal bases of regular wavelets and spline
functions recently constructed by P. Auscher and T. Hyt\"onen, together with
obtaining some crucial lower bounds for regular wavelets, the authors give an
unconditional basis of $H^1_{\rm at}({\mathcal X})$ and several equivalent
characterizations of $H^1_{\rm at}({\mathcal X})$ in terms of wavelets, which
are proved useful.; Comment: 40 pages, submitted. We spilit the article arXiv:1506.05910 into two
papers and this is the first one

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## ‣ Wavelet transform on the torus: a group theoretical approach

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/10/2013
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#Mathematical Physics#Mathematics - Functional Analysis#Mathematics - Representation Theory#81R30, 81R05, 42B05, 42C15

We construct a Continuous Wavelet Transform (CWT) on the torus $\mathbb T^2$
following a group-theoretical approach based on the conformal group $SO(2,2)$.
The Euclidean limit reproduces wavelets on the plane $\mathbb R^2$ with two
dilations, which can be defined through the natural tensor product
representation of usual wavelets on $\mathbb R$. Restricting ourselves to a
single dilation imposes severe conditions for the mother wavelet that can be
overcome by adding extra modular group $SL(2,\mathbb Z)$ transformations, thus
leading to the concept of \emph{modular wavelets}. We define modular-admissible
functions and prove frame conditions.; Comment: 21 pages, 10 figures

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