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## ‣ Regularity of generalized Daubechies wavelets reproducing exponential polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/10/2012
Português

Relevância na Pesquisa

37.727043%

We investigate non-stationary orthogonal wavelets based on a non-stationary
interpolatory subdivision scheme reproducing a given set of exponentials. The
construction is analogous to the construction of Daubechies wavelets using the
subdivision scheme of Deslauriers-Dubuc. The main result is the smoothness of
these Daubechies type wavelets.

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## ‣ Super-wavelets on local fields of positive characteristic

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/11/2015
Português

Relevância na Pesquisa

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The concept of super-wavelet was introduced by Balan, and Han and Larson over
the field of real numbers which has many applications not only in engineering
branches but also in different areas of mathematics. To develop this notion on
local fields having positive characteristic we obtain characterizations of
super-wavelets of finite length as well as Parseval frame multiwavelet sets of
finite order in this setup. Using the group theoretical approach based on coset
representatives, further we establish Shannon type multiwavelet in this
perspective while providing examples of Parseval frame (multi)wavelets and
(Parseval frame) super-wavelets. In addition, we obtain necessary conditions
for decomposable and extendable Parseval frame wavelets associated to Parseval
frame super-wavelets.; Comment: arXiv admin note: text overlap with arXiv:1511.05703

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## ‣ Wavelets in Banach Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

37.737173%

#Mathematics - Functional Analysis#Mathematical Physics#Mathematics - Complex Variables#Mathematics - Representation Theory#Quantum Physics#43A85 (Primary)#32M99, 43A32, 46E10, 47A60, 47A67, 47C99, 81R30,
81S10 (Secondary)

We describe a construction of wavelets (coherent states) in Banach spaces
generated by ``admissible'' group representations. Our main targets are
applications in pure mathematics while connections with quantum mechanics are
mentioned. As an example we consider operator valued Segal-Bargmann type spaces
and the Weyl functional calculus.
Keywords: Wavelets, coherent states, Banach spaces, group representations,
covariant, contravariant (Wick) symbols, Heisenberg group, Segal-Bargmann
spaces, Weyl functional calculus (quantization), second quantization, bosonic
field.; Comment: 37 pages; LaTeX2e; no pictures; 27/07/99: many small corrections

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## ‣ On the usefulness of Meyer wavelets for deconvolution and density estimation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

37.826187%

#Mathematics - Statistics Theory#Mathematics - Functional Analysis#Statistics - Methodology#62G07#42C40, 41A29

The aim of this paper is to show the usefulness of Meyer wavelets for the
classical problem of density estimation and for density deconvolution from
noisy observations. By using such wavelets, the computation of the empirical
wavelet coefficients relies on the fast Fourier transform of the data and on
the fact that Meyer wavelets are band-limited functions. This makes such
estimators very simple to compute and this avoids the problem of evaluating
wavelets at non-dyadic points which is the main drawback of classical
wavelet-based density estimators. Our approach is based on term-by-term
thresholding of the empirical wavelet coefficients with random thresholds
depending on an estimation of the variance of each coefficient. Such estimators
are shown to achieve the same performances of an oracle estimator up to a
logarithmic term. These estimators also achieve near-minimax rates of
convergence over a large class of Besov spaces. A simulation study is proposed
to show the good finite sample performances of the estimator for both problems
of direct density estimation and density deconvolution.

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## ‣ Non-MSF wavelets for the Hardy space H^2(\R)

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/07/2002
Português

Relevância na Pesquisa

37.766992%

We prove three results on wavelets for the Hardy space H^2(\R). All wavelets
constructed so far for H^2(\R) are MSF wavelets. We construct a family of
H^2-wavelets which are not MSF. An equivalence relation on H^2-wavelets is
introduced and it is shown that the corresponding equivalence classes are
non-empty. Finally, we construct a family of H^2-wavelets with Fourier
transform discontinuous at the origin.; Comment: 11 pages

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## ‣ On Filter Banks and Wavelets Based on Chebyshev Polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/11/2014
Português

Relevância na Pesquisa

37.766992%

In this paper we introduce a new family of wavelets, named Chebyshev
wavelets, which are derived from conventional first and second kind Chebyshev
polynomials. Properties of Chebyshev filter banks are investigated, including
orthogonality and perfect reconstruction conditions. Chebyshev wavelets have
compact support, their filters possess good selectivity, but they are not
orthogonal. The convergence of the cascade algorithm of Chebyshev wavelets is
proved by using properties of Markov chains. Computational implementation of
these wavelets and some clear-cut applications are presented. Proposed wavelets
are suitable for signal denoising.; Comment: 18 pages, 6 figures

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## ‣ Higher-Order Properties of Analytic Wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

37.766992%

The influence of higher-order wavelet properties on the analytic wavelet
transform behavior is investigated, and wavelet functions offering advantageous
performance are identified. This is accomplished through detailed investigation
of the generalized Morse wavelets, a two-parameter family of exactly analytic
continuous wavelets. The degree of time/frequency localization, the existence
of a mapping between scale and frequency, and the bias involved in estimating
properties of modulated oscillatory signals, are proposed as important
considerations. Wavelet behavior is found to be strongly impacted by the degree
of asymmetry of the wavelet in both the frequency and the time domain, as
quantified by the third central moments. A particular subset of the generalized
Morse wavelets, recognized as deriving from an inhomogeneous Airy function,
emerge as having particularly desirable properties. These "Airy wavelets"
substantially outperform the only approximately analytic Morlet wavelets for
high time localization. Special cases of the generalized Morse wavelets are
examined, revealing a broad range of behaviors which can be matched to the
characteristics of a signal.; Comment: 15 pages, 6 Postscript figures

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## ‣ $p$-Adic multidimensional wavelets and their application to $p$-adic pseudo-differential operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/12/2006
Português

Relevância na Pesquisa

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

In this paper we study some problems related with the theory of
multidimensional $p$-adic wavelets in connection with the theory of
multidimensional $p$-adic pseudo-differential operators (in the $p$-adic
Lizorkin space). We introduce a new class of $n$-dimensional $p$-adic compactly
supported wavelets. In one-dimensional case this class includes the Kozyrev
$p$-adic wavelets. These wavelets (and their Fourier transforms) form an
orthonormal complete basis in ${\cL}^2(\bQ_p^n)$. A criterion for a
multidimensional $p$-adic wavelet to be an eigenfunction for a
pseudo-differential operator is derived. We prove that these wavelets are
eigenfunctions of the Taibleson fractional operator. Since many $p$-adic models
use pseudo-differential operators (fractional operator), these results can be
intensively used in applications. Moreover, $p$-adic wavelets are used to
construct solutions of linear and {\it semi-linear} pseudo-differential
equations.

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## ‣ Diffusive wavelets on the Spin group

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

37.883525%

#Mathematics - Functional Analysis#Mathematics - Complex Variables#Mathematics - Group Theory#Mathematics - Representation Theory#42C40, 43A75, 43A77, 43A85, 43A90

The first part of this article is devoted to a brief review of the results
about representation theory of the spin group Spin(m) from the point of view of
Clifford analysis. In the second part we are interested in Clifford-valued
functions and wavelets on the sphere. The connection of representations of
Spin(m) and the concept of diffusive wavelets leads naturally to investigations
of a modified diffusion equation on the sphere, that makes use of the Gamma
operator. We will achieve to obtain Clifford-valued diffusion wavelets with
respect to a modified diffusion operator.
Since we are able to characterize all representations of Spin(m) and even to
obtain all eigenvectors of the (by representation) regarded Casimir operator in
representation spaces, it seems appropriate to look at functions on Spin(m)
directly. Concerning this, our aim shall be to formulate eigenfunctions for the
Laplace-Beltrami operator on Spin(m) and give the series expansion of the heat
kernel on Spin(m) in terms of eigenfunctions.; Comment: 28 pages. arXiv admin note: text overlap with arXiv:0809.1408 1
reference added, motivation part rewritten

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## ‣ Uncertainty constants and quasispline wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/12/2009
Português

Relevância na Pesquisa

37.951929%

In 1996 Chui and Wang proved that the uncertainty constants of scaling and
wavelet functions tend to infinity as smoothness of the wavelets grows for a
broad class of wavelets such as Daubechies wavelets and spline wavelets. We
construct a class of new families of wavelets (quasispline wavelets) whose
uncertainty constants tend to those of the Meyer wavelet function used in
construction.; Comment: 27 pages

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## ‣ Continuous Wavelets on Compact Manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/11/2008
Português

Relevância na Pesquisa

37.823914%

#Mathematics - Functional Analysis#Mathematics - Classical Analysis and ODEs#Mathematics - Spectral Theory#42C40, 42B20, 58J40, 58J35, 35P05

Let $\bf M$ be a smooth compact oriented Riemannian manifold, and let
$\Delta_{\bf M}$ be the Laplace-Beltrami operator on ${\bf M}$. Say $0 \neq f
\in \mathcal{S}(\RR^+)$, and that $f(0) = 0$. For $t > 0$, let $K_t(x,y)$
denote the kernel of $f(t^2 \Delta_{\bf M})$. We show that $K_t$ is
well-localized near the diagonal, in the sense that it satisfies estimates akin
to those satisfied by the kernel of the convolution operator $f(t^2\Delta)$ on
$\RR^n$. We define continuous ${\cal S}$-wavelets on ${\bf M}$, in such a
manner that $K_t(x,y)$ satisfies this definition, because of its localization
near the diagonal. Continuous ${\cal S}$-wavelets on ${\bf M}$ are analogous to
continuous wavelets on $\RR^n$ in $\mathcal{S}(\RR^n)$. In particular, we are
able to characterize the H$\ddot{o}$lder continuous functions on ${\bf M}$ by
the size of their continuous ${\mathcal{S}}-$wavelet transforms, for
H$\ddot{o}$lder exponents strictly between 0 and 1. If $\bf M$ is the torus
$\TT^2$ or the sphere $S^2$, and $f(s)=se^{-s}$ (the ``Mexican hat''
situation), we obtain two explicit approximate formulas for $K_t$, one to be
used when $t$ is large, and one to be used when $t$ is small.

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## ‣ A Class of non-MRA Band-limited Wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/08/2001
Português

Relevância na Pesquisa

37.80936%

We give a characterization of a class of band-limited wavelets of
$L^2({\mathbb R})$ and show that none of these wavelets come from a
multiresolution analysis (MRA). For each $n\geq 2$, we construct a subset $S_n$
of ${\mathbb R}$ which is symmetric with respect to the origin. We give
necessary and sufficient conditions on a function $\psi\in L^2({\mathbb R})$
with supp $\hat\psi\subseteq S_n$ to be an orthonormal wavelet. This result
generalizes the characterization of a class of wavelets of E. Hern\'andez and
G. Weiss. The dimension functions associated with these wavelets are also
computed explicitly. Starting from the wavelets we have constructed, we are
able to construct examples of wavelets in each of the equivalence classes of
wavelets defined by E. Weber.; Comment: AMS Latex, 17 pages

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