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## ‣ Nonorthogonal Bases and Phase Decomposition: Properties and Applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In a previous paper [1] it was discussed the viability of functional analysis
using as a basis a couple of generic functions, and hence vectorial
decomposition. Here we complete the paradigm exploiting one of the analysis
methodologies developed there, but applied to phase coordinates, so needing
only one function as a basis. It will be shown that, thanks to the novel
iterative analysis, any function satisfying a rather loose requisite is
ontologically a basis. This in turn generalizes the polar version of the
Fourier theorem to an ample class of nonorthogonal bases. The main advantage of
this generalization is that it inherits some of the properties of the original
Fourier theorem. As a result the new transform has a wide range of applications
and some remarkable consequences. The new tool will be compared with wavelets
and frames. Examples of analysis and reconstruction of functions using the
developed algorithms and generic bases will be given. Some of the properties,
and applications that can promptly benefit from the theory, will be discussed.
The implementation of a matched filter for noise suppression will be used as an
example of the potential of the theory.; Comment: 11 pages

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## ‣ Shape optimisation with multiresolution subdivision surfaces and immersed finite elements

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/10/2015
Português

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#Mathematics - Numerical Analysis#Computer Science - Computational Engineering, Finance, and Science

We develop a new optimisation technique that combines multiresolution
subdivision surfaces for boundary description with immersed finite elements for
the discretisation of the primal and adjoint problems of optimisation. Similar
to wavelets multiresolution surfaces represent the domain boundary using a
coarse control mesh and a sequence of detail vectors. Based on the
multiresolution decomposition efficient and fast algorithms are available for
reconstructing control meshes of varying fineness. During shape optimisation
the vertex coordinates of control meshes are updated using the computed shape
gradient information. By virtue of the multiresolution editing semantics,
updating the coarse control mesh vertex coordinates leads to large-scale
geometry changes and, conversely, updating the fine control mesh coordinates
leads to small-scale geometry changes. In our computations we start by
optimising the coarsest control mesh and refine it each time the cost function
reaches a minimum. This approach effectively prevents the appearance of
non-physical boundary geometry oscillations and control mesh pathologies, like
inverted elements. Independent of the fineness of the control mesh used for
optimisation, on the immersed finite element grid the domain boundary is always
represented with a relatively fine control mesh of fixed resolution. With the
immersed finite element method there is no need to maintain an analysis
suitable domain mesh. In some of the presented two- and three-dimensional
elasticity examples the topology derivative is used for creating new holes
inside the domain.

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## ‣ Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/04/2011
Português

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We propose a class of spherical wavelet bases for the analysis of geophysical
models and forthe tomographic inversion of global seismic data. Its
multiresolution character allows for modeling with an effective spatial
resolution that varies with position within the Earth. Our procedure is
numerically efficient and can be implemented with parallel computing. We
discuss two possible types of discrete wavelet transforms in the angular
dimension of the cubed sphere. We discuss benefits and drawbacks of these
constructions and apply them to analyze the information present in two
published seismic wavespeed models of the mantle, for the statistics and power
of wavelet coefficients across scales. The localization and sparsity properties
of wavelet bases allow finding a sparse solution to inverse problems by
iterative minimization of a combination of the $\ell_2$ norm of data fit and
the $\ell_1$ norm on the wavelet coefficients. By validation with realistic
synthetic experiments we illustrate the likely gains of our new approach in
future inversions of finite-frequency seismic data and show its readiness for
global seismic tomography.; Comment: 43 pages, 11 figures, submitted to Geophysical Journal International

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## ‣ Representations of Cuntz algebras, loop groups and wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/08/2000
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#Mathematics - Functional Analysis#Mathematical Physics#46L60, 47D25, 42A16, 43A65 (Primary)#46L45, 42A65, 41A15
(Secondary)

A theorem of Glimm states that representation theory of an NGCR C*-algebra is
always intractable, and the Cuntz algebra O_N is a case in point. The
equivalence classes of irreducible representations under unitary equivalence
cannot be captured with a Borel cross section. Nonetheless, we prove here that
wavelet representations correspond to equivalence classes of irreducible
representations of O_N, and they are effectively labeled by elements of the
loop group, i.e., the group of measurable functions A:T-->U_N(C). These
representations of O_N are constructed here from an orbit picture analysis of
the infinite-dimensional loop group.; Comment: 6 pages, LaTeX2e "amsproc" class; expanded version of an invited
lecture given by the author at the International Congress on Mathematical
Physics, July 2000 in London

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## ‣ The abstruse meets the applicable: Some aspects of time-frequency analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/07/2006
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The area of Fourier analysis connected to signal processing theory has
undergone a rapid development in the last two decades. The aspect of this
development that has received the most publicity is the theory of wavelets and
their relatives, which involves expansions in terms of sets of functions
generated from a single function by translations and dilations. However, there
has also been much progress in the related area known as \emph{time-frequency
analysis} or \emph{Gabor analysis}, which involves expansions in terms of sets
of functions generated from a single function by translations and modulations.
In this area there are some questions of a concrete and practical nature whose
study reveals connections with aspects of harmonic and functional analysis that
were previously considered quite pure and perhaps rather exotic. In this
expository paper, I give a survey of some of these interactions between the
abstruse and the applicable. It is based on the thematic lectures which I gave
at the Ninth Discussion Meeting on Harmonic Analysis at the Harish-Chandra
Research Institute in Allahabad in October 2005.; Comment: 16 pages

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## ‣ Wavelet subspaces invariant under groups of translation operators

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/07/2006
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We study the action of translation operators on wavelet subspaces. This
action gives rise to an equivalence relation on the set of all wavelets. We
show by explicit construction that each of the associated equivalence classes
is non-empty.; Comment: 8 pages, 2 figures

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## ‣ Tail Behaviour of Mexican Needlets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/07/2013
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In this paper we study the tail behaviour of Mexican needlets, a class of
spherical wavelets introduced by Geller, Mayeli (2009). In particular, we
provide an explicit upper bound depending on the resolution level j and a
parameter s governing the shape of the Mexican needlets.; Comment: 25 pages

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## ‣ On the Limiting Behaviour of Needlets Polyspectra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/07/2013
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#Mathematics - Probability#Astrophysics - Cosmology and Nongalactic Astrophysics#60G60, 62M15, 42C15

This paper provides quantitative Central Limit Theorems for nonlinear
transforms of spherical random fields, in the high frequency limit. The
sequences of fields that we consider are represented as smoothed averages of
spherical Gaussian eigenfunctions and can be viewed as random coefficients from
continuous wavelets/needlets; as such, they are of immediate interest for
spherical data analysis. In particular, we focus on so-called needlets
polyspectra, which are popular tools for nonGaussianity analysis in the
astrophysical community, and on the area of excursion sets. Our results are
based on Stein-Maliavin approximations for nonlinear transforms of Gaussian
fields, and on an explicit derivation on the high-frequency limit of their
variances, which may have some independent interest.

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## ‣ Multiscale Geometric Methods for Data Sets II: Geometric Multi-Resolution Analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Metric Geometry#Computer Science - Data Structures and Algorithms#Statistics - Machine Learning

Data sets are often modeled as point clouds in $R^D$, for $D$ large. It is
often assumed that the data has some interesting low-dimensional structure, for
example that of a $d$-dimensional manifold $M$, with $d$ much smaller than $D$.
When $M$ is simply a linear subspace, one may exploit this assumption for
encoding efficiently the data by projecting onto a dictionary of $d$ vectors in
$R^D$ (for example found by SVD), at a cost $(n+D)d$ for $n$ data points. When
$M$ is nonlinear, there are no "explicit" constructions of dictionaries that
achieve a similar efficiency: typically one uses either random dictionaries, or
dictionaries obtained by black-box optimization. In this paper we construct
data-dependent multi-scale dictionaries that aim at efficient encoding and
manipulating of the data. Their construction is fast, and so are the algorithms
that map data points to dictionary coefficients and vice versa. In addition,
data points are guaranteed to have a sparse representation in terms of the
dictionary. We think of dictionaries as the analogue of wavelets, but for
approximating point clouds rather than functions.; Comment: Re-formatted using AMS style

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## ‣ Constructing pairs of dual bandlimited frame wavelets in $L^2(\mathbb{R}^n)$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/09/2010
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Given a real, expansive dilation matrix we prove that any bandlimited
function $\psi \in L^2(\mathbb{R}^n)$, for which the dilations of its Fourier
transform form a partition of unity, generates a wavelet frame for certain
translation lattices. Moreover, there exists a dual wavelet frame generated by
a finite linear combination of dilations of $\psi$ with explicitly given
coefficients. The result allows a simple construction procedure for pairs of
dual wavelet frames whose generators have compact support in the Fourier domain
and desired time localization. The construction relies on a technical condition
on $\psi$, and we exhibit a general class of function satisfying this
condition.; Comment: 21 pages, 6 figures

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## ‣ Best approximation with wavelets in weighted Orlicz spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/11/2009
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Democracy functions of wavelet admissible bases are computed for weighted
Orlicz Spaces in terms of its fundamental function. In particular, we prove
that these bases are greedy if and only if the Orlicz space is a Lebesgue
space. Also, sharp embeddings for the approximation spaces are given in terms
of weighted discrete Lorentz spaces. For Lebesgue spaces the approximation
spaces are identified with weighted Besov spaces.; Comment: 22 pages with references

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## ‣ Groupoid Methods in Wavelet Analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/09/2007
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We describe how the Deaconu-Renault groupoid may be used in the study of
wavelets and fractals.; Comment: To appear in "Group representations, ergodic theory, and mathematical
physics: A tribute to George W. Mackey", the proceedings of the AMS special
session dedicated to the memory of George W. Mackey at the January 2007 AMS
meeting

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## ‣ Besov Spaces and Frames on Compact Manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We show that one can characterize the Besov spaces on a smooth compact
oriented Riemannian manifold, for the full range of indices, through a
knowledge of the size of frame coefficients, using the frames we have
constructed in [8].; Comment: 27 pages. This paper is a continuation of our article ``Nearly Tight
Frames and Space-Frequency Analysis on Compact Manifolds''. Keywords and
phrases: Wavelets, Frames, Spectral Theory, Besov Spaces, Manifolds,
Pseudodifferential Operators

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## ‣ The role of transfer operators and shifts in the study of fractals: encoding-models, analysis and geometry, commutative and non-commutative

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/02/2013
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We study a class of dynamical systems in $L^2$ spaces of infinite products
$X$. Fix a compact Hausdorff space $B$. Our setting encompasses such cases when
the dynamics on $X = B^\bn$ is determined by the one-sided shift in $X$, and by
a given transition-operator $R$. Our results apply to any positive operator $R$
in $C(B)$ such that $R1 = 1$. From this we obtain induced measures $\Sigma$ on
$X$, and we study spectral theory in the associated $L^2(X,\Sigma)$.
For the second class of dynamics, we introduce a fixed endomorphism $r$ in
the base space $B$, and specialize to the induced solenoid $\Sol(r)$. The
solenoid $\Sol(r)$ is then naturally embedded in $X = B^\bn$, and $r$ induces
an automorphism in $\Sol(r)$. The induced systems will then live in
$L^2(\Sol(r), \Sigma)$.
The applications include wavelet analysis, both in the classical setting of
$\br^n$, and Cantor-wavelets in the setting of fractals induced by affine
iterated function systems (IFS). But our solenoid analysis includes such
hyperbolic systems as the Smale-Williams attractor, with the endomorphism $r$
there prescribed to preserve a foliation by meridional disks. And our setting
includes the study of Julia set-attractors in complex dynamics.

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## ‣ Wavelets for iterated function systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/10/2009
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We construct a wavelet and a generalised Fourier basis with respect to some
fractal measures given by one-dimensional iterated function systems. In this
paper we will not assume that these systems are given by linear contractions
generalising in this way some previous work of Jorgensen and Dutkay to the
non-linear setting.; Comment: 17 pages, 3 figures

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## ‣ Pseudodifferential p-adic vector fields and pseudodifferentiation of a composite p-adic function

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/05/2011
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We discuss transformation of p-adic pseudodifferential operators (in the
one-dimensional and multidimensional cases) with respect to p-adic maps which
correspond to automorphisms of the tree of balls in the corresponding p-adic
spaces.
In the dimension one we find a rule of transformation for pseudodifferential
operators. In particular we find the formula of pseudodifferentiation of a
composite function with respect to the Vladimirov p-adic fractional
differentiation operator. We describe the frame of wavelets for the group of
parabolic automorphisms of the tree of balls in the p-adic field.
In many dimensions we introduce the group of mod p-affine transformations,
the family of pseudodifferential operators corresponding to
pseudodifferentiation along vector fields on the tree of balls in p-adic
miltidimensional space and obtain a rule of transformation of the introduced
pseudodifferential operators with respect to mod p-affine transformations.

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## ‣ On adaptive wavelet estimation of a class of weighted densities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We investigate the estimation of a weighted density taking the form
$g=w(F)f$, where $f$ denotes an unknown density, $F$ the associated
distribution function and $w$ is a known (non-negative) weight. Such a class
encompasses many examples, including those arising in order statistics or when
$g$ is related to the maximum or the minimum of $N$ (random or fixed)
independent and identically distributed (\iid) random variables. We here
construct a new adaptive non-parametric estimator for $g$ based on a plug-in
approach and the wavelets methodology. For a wide class of models, we prove
that it attains fast rates of convergence under the $\mathbb{L}_p$ risk with
$p\ge 1$ (not only for $p = 2$ corresponding to the mean integrated squared
error) over Besov balls. The theoretical findings are illustrated through
several simulations.

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## ‣ Generalized pointwise H\"older spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/07/2013
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In this paper, we introduce a generalization of the pointwise H\"older
spaces. We give alternative definitions of these spaces, look at their
relationship with the wavelets and introduce a notion of generalized H\"older
exponent.; Comment: 18 pages

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## ‣ Pairs of Frequency-based Nonhomogeneous Dual Wavelet Frames in the Distribution Space

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In this paper, we study nonhomogeneous wavelet systems which have close
relations to the fast wavelet transform and homogeneous wavelet systems. We
introduce and characterize a pair of frequency-based nonhomogeneous dual
wavelet frames in the distribution space; the proposed notion enables us to
completely separate the perfect reconstruction property of a wavelet system
from its stability property in function spaces. The results in this paper lead
to a natural explanation for the oblique extension principle, which has been
widely used to construct dual wavelet frames from refinable functions, without
any a priori condition on the generating wavelet functions and refinable
functions. A nonhomogeneous wavelet system, which is not necessarily derived
from refinable functions via a multiresolution analysis, not only has a natural
multiresolution-like structure that is closely linked to the fast wavelet
transform, but also plays a basic role in understanding many aspects of wavelet
theory. To illustrate the flexibility and generality of the approach in this
paper, we further extend our results to nonstationary wavelets with real
dilation factors and to nonstationary wavelet filter banks having the perfect
reconstruction property.

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## ‣ Adaptive density estimation for directional data using needlets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/07/2008
Português

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This paper is concerned with density estimation of directional data on the
sphere. We introduce a procedure based on thresholding on a new type of
spherical wavelets called {\it needlets}. We establish a minimax result and
prove its optimality. We are motivated by astrophysical applications, in
particular in connection with the analysis of ultra high energy cosmic rays.; Comment: 30 pages, 9 figures

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