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## ‣ Análisis con Wavelets de alteraciones electrocardiográficas en pacientes Chagásticos crónicos

Riveros Sanabria, Fernando
Tipo: masterThesis; Tesis de Maestría; acceptedVersion
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## ‣ Analysis with Wavelets of electrocardiographic alterations in chronic Chagas patients

Riveros Sanabria, Fernando; Villegas Guti??rrez, Jairo Alberto
Fonte: HIKARI Ltd, www.m-hikari.com; Grupo de Investigaci??n An??lisis Funcional y Aplicaciones; Escuela de Ciencias y Humanidades Publicador: HIKARI Ltd, www.m-hikari.com; Grupo de Investigaci??n An??lisis Funcional y Aplicaciones; Escuela de Ciencias y Humanidades
Tipo: article; Art??culo; publishedVersion
Português
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In this work, an algorithm was created to detect the characteristics of the electrocardiographic signal (ECG) of chronic Chagas patients from the State of Meta, Colombia and records of healthy people, using the Daubechies 5 wavelet (db 5), as an alternative way in the Fourier analysis for this type of signals -- A database of 32 ECG records was created using a single channel with a 16-bits high resolution polygraph -- This algorithm allowed the identification of the most important characteristics of each ECG record of patients with the disease, measuring the intervals and amplitudes of the waves and heart rate with an accuracy greater than 91%, becoming a tool for a better diagnosis of the symptoms of chronic Chagas disease

## ‣ Use of operator algebras in the analysis of measures from wavelets and iterated function systems

Jorgensen, Palle E. T.
Tipo: Artigo de Revista Científica
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In this paper, we show how a class of operators used in the analysis of measures from wavelets and iterated function systems may be understood from a special family of representations of Cuntz algebras.

## ‣ A Construction of Biorthogonal Wavelets With a Compact Operator

Ersoy, Mehmet
Tipo: Artigo de Revista Científica
Português
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We present a construction of biorthogonal wavelets using a compact operator which allows to preserve or increase some properties: regularity/vanishing moments, parity, compact supported. We build then a simple algorithm which computes new filters.

## ‣ Diffusive wavelets on groups and homogeneous spaces

Ebert, Svend; Wirth, Jens
Tipo: Artigo de Revista Científica
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The aim of this exposition is to explain basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by Peter-Weyl decomposition of $L^2(G)$ for a compact Lie group $G$. After developing a general concept for compact groups and their homogeneous spaces we give concrete examples for tori -which reflect the situation on $R^n$- and for spheres $S^2$ and $S^3$.; Comment: 20 pages, 3 figures

## ‣ Scattering Calculations with Wavelets

Kessler, B. M.; Payne, G. L.; Polyzou, W. N.
Tipo: Artigo de Revista Científica
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We show that the use of wavelet bases for solving the momentum-space scattering integral equation leads to sparse matrices which can simplify the solution. Wavelet bases are applied to calculate the K-matrix for nucleon-nucleon scattering with the s-wave Malfliet-Tjon V potential. We introduce a new method, which uses special properties of the wavelets, for evaluating the singular part of the integral. Analysis of this test problem indicates that a significant reduction in computational size can be achieved for realistic few-body scattering problems.; Comment: 26 pages, Latex, 6 eps figures

## ‣ Wavelets on Irregular Grids with Arbitrary Dilation Matrices, and Frames Atoms for L^2(R^d)

Aldroubi, Akram; Cabrelli, Carlos; Molter, Ursula M.
Tipo: Artigo de Revista Científica
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In this article, we develop a general method for constructing wavelets {|det A_j|^{1/2} g(A_jx-x_{j,k}): j in J, k in K}, on irregular lattices of the form X={x_{j,k} in R^d: j in J, k in K}, and with an arbitrary countable family of invertible dxd matrices {A_j in GL_d(R): j in J} that do not necessarily have a group structure. This wavelet construction is a particular case of general atomic frame decompositions of L^2(R^d) developed in this article, that allow other time frequency decompositions such as non-harmonic Gabor frames with non-uniform covering of the Euclidean space R^d. Possible applications include image and video compression, speech coding, image and digital data transmission, image analysis, estimations and detection, and seismology.; Comment: 23 pages, 3 figures, some small correction after publication

## ‣ Locally Supported Wavelets for the Separation of Spherical Vector Fields with Respect to their Sources

Gerhards, Christian
Tipo: Artigo de Revista Científica
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We provide a space domain oriented separation of magnetic fields into parts generated by sources in the exterior and sources in the interior of a given sphere. The separation itself is well-known in geomagnetic modeling, usually in terms of a spherical harmonic analysis or a wavelet analysis that is spherical harmonic based. In contrast to these frequency oriented methods, we use a more spatially oriented approach in this paper. We derive integral representations with explicitly known convolution kernels. Regularizing these singular kernels allows a multiscale representation of the internal and external contributions to the magnetic field with locally supported wavelets. This representation is applied to a set of CHAMP data for crustal field modeling.

## ‣ Application of second generation wavelets to blind spherical deconvolution

Vareschi, Thomas
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We adress the problem of spherical deconvolution in a non parametric statistical framework, where both the signal and the operator kernel are subject to error measurements. After a preliminary treatment of the kernel, we apply a thresholding procedure to the signal in a second generation wavelet basis. Under standard assumptions on the kernel, we study the theoritical performance of the resulting algorithm in terms of $L^p$ losses ($p\geq 1$) on Besov spaces on the sphere. We hereby extend the application of second generation spherical wavelets to the blind deconvolution framework. The procedure is furthermore adaptive with regard both to the target function sparsity and smoothness, and the kernel blurring effect. We end with the study of a concrete example, putting into evidence the improvement of our procedure on the recent blockwise-SVD algorithm.

## ‣ Simplified vanishing moment criteria for wavelets over general dilation groups, with applications to abelian and shearlet dilation groups

Führ, Hartmut; Tousi, Reihaneh Raisi
Tipo: Artigo de Revista Científica
Português
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We consider the coorbit theory associated to general continuous wavelet transforms arising from a square-integrable, irreducible quasi-regular representation of a semidirect product group $G = \mathbb{R}^d \rtimes H$. The existence of coorbit spaces for this very general setting has been recently established, together with concrete vanishing moment criteria for analyzing vectors and atoms that can be used in the coorbit scheme. These criteria depend on fairly technical assumptions on the dual action of the dilation group, and it is one of the chief purposes of this paper to considerably simplify these assumptions. We then proceed to verify the assumptions for large classes of dilation groups, in particular for all abelian dilation groups, as well as a class called {\em generalized shearlet dilation groups}, containing and extending all known examples of shearlet dilation groups employed in dimensions two and higher. We explain how these groups can be systematically constructed from certain commutative associative algebras of the same dimension, and give a full list, up to conjugacy, of shearing groups in dimensions three and four. In the latter case, three previously unknown groups arise. As a result the existence of Banach frames consisting of compactly supported wavelets...

## ‣ Spectral Models for Orthonormal Wavelets and Multiresolution Analysis of $L^2({\mathbb R})$

Gómez-Cubillo, F.; Suchanecki, Z.
Tipo: Artigo de Revista Científica
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Spectral representations of the dilation and translation operators on $L^2({\mathbb R})$ are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions defined on the functional spectral spaces. The approach is useful for computational purposes.; Comment: 26 pages

## ‣ Density estimation on the rotation group using diffusive wavelets

Bihan, Nicolas Le; Flamant, Julien; Manton, Jonathan H.
Tipo: Artigo de Revista Científica
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This paper considers the problem of estimating probability density functions on the rotation group $SO(3)$. Two distinct approaches are proposed, one based on characteristic functions and the other on wavelets using the heat kernel. Expressions are derived for their Mean Integrated Squared Errors. The performance of the estimators is studied numerically and compared with the performance of an existing technique using the De La Vall\'ee Poussin kernel estimator. The heat-kernel wavelet approach appears to offer the best convergence, with faster convergence to the optimal bound and guaranteed positivity of the estimated probability density function.

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

Jorgensen, Palle E. T.
Tipo: Artigo de Revista Científica
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We show that compactly supported wavelets in L^2(R) of scale N may be effectively parameterized with a finite set of spin vectors in C^N, and conversely that every set of spin vectors corresponds to a wavelet. The characterization is given in terms of irreducible representations of orthogonality relations defined from multiresolution wavelet filters.; Comment: 10 or 11 pages, SPIE Technical Conference, Wavelet Applications in Signal and Image Processing VIII

## ‣ Orthonormal dilations of Parseval wavelets

Dutkay, Dorin Ervin; Han, Deguang; Picioroaga, Gabriel; Sun, Qiyu
Tipo: Artigo de Revista Científica
Português
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We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group $$BS(1,2)=< u,t | utu^{-1}=t^2>.$$ We give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics. We show that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics. We give concrete examples of Parseval wavelets for which we compute the orthonormal dilations in detail; we show that there are examples of Parseval wavelet sets which have infinitely many non-isomorphic orthonormal dilations.; Comment: v2, improved introduction according to the referee's suggestions, corrected some typos. Accepted for Mathematische Annalen

## ‣ Multidimensional p-adic wavelets for the deformed metric

Albeverio, S.; Kozyrev, S. V.
Tipo: Artigo de Revista Científica
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The approach to p-adic wavelet theory from the point of view of representation theory is discussed. p-Adic wavelet frames can be constructed as orbits of some p-adic groups of transformations. These groups are automorphisms of the tree of balls in the p-adic space. In the present paper we consider deformations of the standard p-adic metric in many dimensions and construct some corresponding groups of transformations. We build several examples of p-adic wavelet bases. We show that the constructed wavelets are eigenvectors of some pseudodifferential operators.

## ‣ Direct limits, multiresolution analyses, and wavelets

Baggett, Lawrence W.; Larsen, Nadia S.; Packer, Judith A.; Raeburn, Iain; Ramsay, Arlan
Tipo: Artigo de Revista Científica
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A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on $L^2(\R^n)$, the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids.; Comment: 23 pages including bibligraphy

## ‣ Bilinear Decompositions of Products of Hardy and Lipschitz or BMO Spaces Through Wavelets

Cao, Jun; Ky, Luong Dang; Yang, Dachun
Tipo: Artigo de Revista Científica
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The aim of this article is to give a complete solution to the problem of the bilinear decompositions of the products of some Hardy spaces $H^p(\mathbb{R}^n)$ and their duals in the case when $p<1$ and near to $1$, via wavelets, paraproducts and the theory of bilinear Calder\'on-Zygmund operators. Precisely, the authors establish the bilinear decompositions of the product spaces $H^p(\mathbb{R}^n)\times\dot\Lambda_{\alpha} (\mathbb{R}^n)$ and $H^p(\mathbb{R}^n)\times\Lambda_{\alpha}(\mathbb{R}^n)$, where, for all $p\in(\frac{n}{n+1},\,1)$ and $\alpha:=n(\frac{1}{p}-1)$, $H^p(\mathbb{R}^n)$ denotes the classical real Hardy space, and $\dot\Lambda_{\alpha}$ and $\Lambda_{\alpha}$ denote, respectively, the homogeneous and the inhomogeneous Lipschitz spaces. Sharpness of these two bilinear decompositions are also proved. Moreover, the authors also give the corresponding bilinear decompositions of the associated local product spaces $h^1(\mathbb{R}^n)\times{\mathop\mathrm{bmo}}(\mathbb{R}^n)$ and $h^p(\mathbb{R}^n)\times\Lambda_{\alpha}(\mathbb{R}^n)$ with $p\in(\frac{n}{n+1},\,1)$ and $\alpha:=n(\frac{1}{p}-1)$, where, for all $p\in(\frac{n}{n+1},\,1]$, $h^p(\mathbb{R}^n)$ denotes the local Hardy space and ${\mathop\mathrm{bmo}}(\mathbb{R}^n)$ the local BMO space in the sense of D. Goldberg. As an application...

## ‣ Spaces of Analytical Functions and Wavelets--Lecture Notes

Tipo: Artigo de Revista Científica
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This is (raw) lecture notes of the course read on 6th European intensive course on Complex Analysis (Coimbra, Portugal) in 2000. Our purpose is to describe a general framework for generalizations of the complex analysis. As a consequence a classification scheme for different generalizations is obtained. The framework is based on wavelets (coherent states) in Banach spaces generated by admissible'' group representations. Reduced wavelet transform allows naturally describe in abstract term main objects of an analytical function theory: the Cauchy integral formula, the Hardy and Bergman spaces, the Cauchy-Riemann equation, and the Taylor expansion. Among considered examples are classical analytical function theories (one complex variables, several complex variables, Clifford analysis, Segal-Bargmann space) as well as new function theories which were developed within our framework (function theory of hyperbolic type, Clifford version of Segal-Bargmann space). We also briefly discuss applications to the operator theory (functional calculus) and quantum mechanics.; Comment: LaTeX, pages 92, two PS picture

## ‣ Products of Functions in ${\mathop\mathrm{BMO}}({\mathcal X})$ and $H^1_{\rm at}({\mathcal X})$ via Wavelets over Spaces of Homogeneous Type

Fu, Xing; Yang, Dachun; Liang, Yiyu
Tipo: Artigo de Revista Científica
Português
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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, the authors prove that the product $f\times g$ of $f\in H^1_{\rm at}({\mathcal X})$ and $g\in\mathop\mathrm{BMO}({\mathcal X})$, viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $H^1_{\rm at}({\mathcal X})\times\mathop\mathrm{BMO}({\mathcal X})$ into $L^1({\mathcal X})$ and from $H^1_{\rm at}({\mathcal X})\times\mathop\mathrm{BMO}({\mathcal X})$ into $H^{\log}({\mathcal X})$, which affirmatively confirms the conjecture suggested by A. Bonami and F. Bernicot (This conjecture was presented by L. D. Ky in [J. Math. Anal. Appl. 425 (2015), 807-817]).; Comment: 66 pages, Submitted

## ‣ From filters to wavelets via direct limits

We present a new proof of a theorem of Mallat which describes a construction of wavelets starting from a quadrature mirror filter. Our main innovation is to show how the scaling function associated to the filter can be used to identify a certain direct limit of Hilbert spaces with $L^2(\R)$ in such a way that one can immediately identify the wavelet basis. Our arguments also use a pair of isometries introduced by Bratteli and Jorgensen, and exploit the geometry inherent in the Cuntz relations satisfied by these isometries.; Comment: 6 pages, to appear in GPOTS 2005 proceedings