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‣ Paley-Wiener description of K-spherical Besov spaces on the Heisenberg group

Mayeli, Azita
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 19/11/2011 Português
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We characterize the Besov spaces associated to the Gelfand pairs on the Heisenberg group. The characterization is given in terms of bandlimited wavelet coefficients where the bandlimitedness is introduced using spherical Fourier transform. To obtain these results we develop an approach to the characterization of Besov spaces in abstract Hilbert spaces through compactly supported admissible functions.; Comment: Keywords: Besov spaces, Paley-Wiener spaces, Gelfand pair, the Heisenberg group, spherical transform, wavelets

‣ Mexican Hat Wavelet on the Heisenberg Group

Mayeli, Azita
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 23/05/2007 Português
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In this article wavelets (admissible vectors) on the Heisenberg group are studied from the point of view of Calderon's formula. Further we shall show that for the class of Schwartz functions the Calderon admissibility condition is equivalent to the usual admissibility property which will be introduced in this work. Furthermore motivated by a well-known example on the real line, the Mexican-Hat wavelet, we demonstrate the existence and construction of an analogous wavelet on the Heisenberg Lie group with 2 vanishing moments, which together with all of its derivatives has Gaussian decay.; Comment: 8 pages, no figures

‣ A Haar-type Approximation and a New Numerical Schema for the Korteweg-de Vries Equation

Baggett, Jason; Bastille, Odile; Rybkin, Alexei
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 18/07/2011 Português
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We discuss a new numerical schema for solving the initial value problem for the Korteweg-de Vries equation for large times. Our approach is based upon the Inverse Scattering Transform that reduces the problem to calculating the reflection coefficient of the corresponding Schr\"odinger equation. Using a step-like approximation of the initial profile and a fragmentation principle for the scattering data, we obtain an explicit recursion formula for computing the reflection coefficient, yielding a high resolution KdV solver. We also discuss some generalizations of this algorithm and how it might be improved by using Haar and other wavelets.

‣ Certain representations of the Cuntz relations, and a question on wavelets decompositions

Jorgensen, Palle E. T.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We compute the Coifman-Meyer-Wickerhauser measure $\mu$ for certain families of quadrature mirror filters (QMFs), and we establish that for a subclass of QMFs, $\mu$ contains a fractal scale. In particular, these measures $\mu$ are not in the Lebesgue class.; Comment: v.2 has a new title and additional material in the introduction. Prepared using the amsproc.cls document class

‣ Gabor fields and wavelet sets for the Heisenberg group

Currey, Bradley; Mayeli, Azita
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We study singly-generated wavelet systems on $\Bbb R^2$ that are naturally associated with rank-one wavelet systems on the Heisenberg group $N$. We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset $I$ of the dual of $N$, we give an explicit construction for Parseval frame wavelets that are associated with $I$. We say that $g\in L^2(I\times \Bbb R)$ is Gabor field over $I$ if, for a.e. $\lambda \in I$, $|\lambda|^{1/2} g(\lambda,\cdot)$ is the Gabor generator of a Parseval frame for $L^2(\Bbb R)$, and that $I$ is a Heisenberg wavelet set if every Gabor field over $I$ is a Parseval frame (mother-)wavelet for $L^2(\Bbb R^2)$. We then show that $I$ is a Heisenberg wavelet set if and only if $I$ is both translation congruent with a subset of the unit interval and dilation congruent with the Shannon set.

‣ Interpolation maps and congruence domains for wavelet sets

Zhang, Xiaofei; Larson, David R.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 28/10/2007 Português
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It is proven that if an interpolation map between two wavelet sets preserves the union of the sets, then the pair must be an interpolation pair. We also construct an example of a pair of wavelet sets for which the congruence domains of the associated interpolation map and its inverse are equal, and yet the pair is not an interpolation pair. The first result solves affirmatively a problem that the second author had posed several years ago, and the second result solves an intriguing problem of D. Han. The key to this counterexample is a special technical lemma on constructing wavelet sets. Several other applications of this result are also given. In addition, some problems are posed. We also take the opportunity to give some general exposition on wavelet sets and operator-theoretic interpolation of wavelets.; Comment: 21 pages, 2 figures

‣ A characterization of product BMO by commutators

Lacey, Michael; Ferguson, Sarah
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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Let b be a function on the plane. Let H_j, j=1,2, be the Hilbert transform acting on the j-th coordinate on the plane. We show that the operator norm of the double commutator [[ M_b, H_1], H_2] is equivalent to the Chang-Fefferman BMO norm of b. Here, M_b denotes the operator which is multiplication by b. This result extends a well known theorem of Nehari on weak factorization in the Hardy space H^1 to the same theorem on H^1 of a product domain. The product setting is more delicate because of the presence of a two parameter family of dilations. The method of proof depends upon (a) A dyadic decomposition of product BMO by wavelets (b) a prior estimate of Ferguson and Sadosky involving rectangular BMO (c) and a careful control of certain measures related to those of Carleson.; Comment: 21 pages. To appear in Acta Math. Appendix corrected

‣ Low-pass filters and representations of the Baumslag Solitar group

Dutkay, Dorin Ervin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We analyze representations of solvable Baumslag Solitar group that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.

‣ $p$-Adic Haar multiresolution analysis and pseudo-differential operators

Shelkovich, V. M.; Skopina, M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/05/2007 Português
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The notion of {\em $p$-adic multiresolution analysis (MRA)} is introduced. We discuss a ``natural'' refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of $p$ characteristic functions of mutually disjoint discs of radius $p^{-1}$. This refinement equation generates a MRA. The case $p=2$ is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ${\cL}^2(\bQ_2)$ generated by the same Haar MRA. All of these bases are described. We also constructed multidimensional 2-adic Haar orthonormal bases for ${\cL}^2(\bQ_2^n)$ by means of the tensor product of one-dimensional MRAs. A criterion for a multidimensional $p$-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our bases in applications.

‣ Paraproducts and Products of functions in $BMO(\mathbb R^n)$ and $H^1(\mathbb R^n)$ through wavelets

Bonami, Aline; Grellier, Sandrine; Ky, Luong Dang
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/03/2011 Português
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In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may be written as the sum of two continuous bilinear operators, one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$, the other one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of Hardy-Orlicz space denoted by $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the set of distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies $$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathcal Mf(x)|)}dx <\infty.$$ The two bilinear operators can be defined in terms of paraproducts. As a consequence, we find an endpoint estimate involving the space $\H^{\log}(\bR^n)$ for the $\div$-$\curl$ lemma.

‣ The near shift-invariance of the dual-tree complex wavelet transform revisited

Barri, Adriaan; Dooms, Ann; Schelkens, Peter
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 30/04/2013 Português
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The dual-tree complex wavelet transform (DTCWT) is an enhancement of the conventional discrete wavelet transform (DWT) due to a higher degree of shift-invariance and a greater directional selectivity, finding its applications in signal and image processing. This paper presents a quantitative proof of the superiority of the DTCWT over the DWT in case of modulated wavelets.; Comment: 15 pages

‣ Reproducing subgroups of $Sp(2,\mathbb{R})$. Part I: algebraic classification

Alberti, Giovanni S.; Balletti, Luca; De Mari, Filippo; De Vito, Ernesto
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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We classify the connected Lie subgroups of the symplectic group $Sp(2,\mathbb{R})$ whose elements are matrices in block lower triangular form. The classification is up to conjugation within $Sp(2,\mathbb{R})$. Their study is motivated by the need of a unified approach to continuous 2D signal analyses, as those provided by wavelets and shearlets.; Comment: 26 pages

‣ p-Adic refinable functions and MRA-based wavelets

Khrennikov, A. Yu.; Shelkovich, V. M.; Skopina, M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 18/11/2007 Português
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We described a wide class of $p$-adic refinable equations generating $p$-adic multiresolution analysis. A method for the construction of $p$-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an example, which gives a new 3-adic wavelet basis. Another realization leads to the $p$-adic Haar bases which were known before.; Comment: 10 pages

‣ On alternative wavelet reconstruction formula: a case study of approximate wavelets

Lebedeva, Elena A.; Postnikov, Eugene B.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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The application of the continuous wavelet transform to study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies due to the admissibility condition. We propose an alternative reconstruction formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated. The case of the transform with the standard Morlet wavelet, which is an important example of such analyzing functions, is discussed.; Comment: 6 pages

‣ A Discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmetric spaces

Pesenson, Isaac
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/04/2011 Português
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Let $f$ be a Paley-Wiener function in the space $L_{2}(X)$, where $X$ is a symmetric space of noncompact type. It is shown that by using the values of $f$ on a sufficiently dense and separated set of points of $X$ one can give an exact formula for the Helgason-Fourier transform of $f$. In order to find a discrete approximation to the Helgason-Fourier transform of a function from a Besov space on $X$ we develop an approximation theory by Paley-Wiener functions in $L_{2}(X)$.; Comment: Radon transforms, geometry, and wavelets, 231-247, Contemp. Math., 464, Amer. Math. Soc., Providence, RI, 2008

‣ Almost diagonal matrices and Besov-type spaces based on wavelet expansions

Weimar, Markus
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 20/08/2014 Português
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This paper is concerned with problems in the context of the theoretical foundation of adaptive (wavelet) algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of $d$-dimensional manifolds and investigate some analytical properties (such as, e.g., embeddings and best $n$-term approximation rates) of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $\Psi$ on domains or manifolds $\Gamma$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$, provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity (compared to the smoothness parameter $\alpha$ of the space). For this purpose...

‣ Subsampling needlet coefficients on the sphere

Baldi, P.; Kerkyacharian, G.; Marinucci, D.; Picard, D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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In a recent paper, we analyzed the properties of a new kind of spherical wavelets (called needlets) for statistical inference procedures on spherical random fields; the investigation was mainly motivated by applications to cosmological data. In the present work, we exploit the asymptotic uncorrelation of random needlet coefficients at fixed angular distances to construct subsampling statistics evaluated on Voronoi cells on the sphere. We illustrate how such statistics can be used for isotropy tests and for bootstrap estimation of nuisance parameters, even when a single realization of the spherical random field is observed. The asymptotic theory is developed in detail in the high resolution sense.; Comment: Published in at http://dx.doi.org/10.3150/08-BEJ164 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

‣ Nonlinear Inversion from Partial EIT Data: Computational Experiments

Hamilton, Sarah Jane; Siltanen, Samuli
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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Electrical impedance tomography (EIT) is a non-invasive imaging method in which an unknown physical body is probed with electric currents applied on the boundary, and the internal conductivity distribution is recovered from the measured boundary voltage data. The reconstruction task is a nonlinear and ill-posed inverse problem, whose solution calls for special regularized algorithms, such as D-bar methods which are based on complex geometrical optics solutions (CGOs). In many applications of EIT, such as monitoring the heart and lungs of unconscious intensive care patients or locating the focus of an epileptic seizure, data acquisition on the entire boundary of the body is impractical, restricting the boundary area available for EIT measurements. An extension of the D-bar method to the case when data is collected only on a subset of the boundary is studied by computational simulation. The approach is based on solving a boundary integral equation for the traces of the CGOs using localized basis functions (Haar wavelets). The numerical evidence suggests that the D-bar method can be applied to partial-boundary data in dimension two and that the traces of the partial data CGOs approximate the full data CGO solutions on the available portion of the boundary...

‣ Geometric Separation by Single-Pass Alternating Thresholding

Kutyniok, Gitta
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/04/2012 Português
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Modern data is customarily of multimodal nature, and analysis tasks typically require separation into the single components. Although a highly ill-posed problem, the morphological difference of these components sometimes allow a very precise separation such as, for instance, in neurobiological imaging a separation into spines (pointlike structures) and dendrites (curvilinear structures). Recently, applied harmonic analysis introduced powerful methodologies to achieve this task, exploiting specifically designed representation systems in which the components are sparsely representable, combined with either performing $\ell_1$ minimization or thresholding on the combined dictionary. In this paper we provide a thorough theoretical study of the separation of a distributional model situation of point- and curvilinear singularities exploiting a surprisingly simple single-pass alternating thresholding method applied to the two complementary frames: wavelets and curvelets. Utilizing the fact that the coefficients are clustered geometrically, thereby exhibiting clustered/geometric sparsity in the chosen frames, we prove that at sufficiently fine scales arbitrarily precise separation is possible. Even more surprising, it turns out that the thresholding index sets converge to the wavefront sets of the point- and curvilinear singularities in phase space and that those wavefront sets are perfectly separated by the thresholding procedure. Main ingredients of our analysis are the novel notion of cluster coherence and clustered/geometric sparsity as well as a microlocal analysis viewpoint.; Comment: 35 pages...

‣ Optimal Decompositions of Translations of $L^{2}$-functions

Jorgensen, Palle E. T.; Song, Myung-Sin
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 30/11/2007 Português
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In this paper we offer a computational approach to the spectral function for a finite family of commuting operators, and give applications. Motivated by questions in wavelets and in signal processing, we study a problem about spectral concentration of integral translations of functions in the Hilbert space $L^{2}(\mathbb{R}^{n})$. Our approach applies more generally to families of $n$ arbitrary commuting unitary operators in a complex Hilbert space $\mathcal{H}$, or equivalent the spectral theory of a unitary representation $U$ of the rank-$n$ lattice $\mathbb{Z}^{n}$ in $\mathbb{R}^{n}$. Starting with a non-zero vector $\psi \in \mathcal{H}$, we look for relations among the vectors in the cyclic subspace in $\mathcal{H}$ generated by $\psi$. Since these vectors $\{U(k)\psi | k \in \mathbb{Z}^{n}\}$ involve infinite ``linear combinations," the problem arises of giving geometric characterizations of these non-trivial linear relations. A special case of the problem arose initially in work of Kolmogorov under the name $L^{2}$-independence. This refers to \textit{infinite} linear combinations of integral translates of a fixed function with $l^{2}$-coefficients. While we were motivated by the study of translation operators arising in wavelet and frame theory...