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## ‣ Sharp growth estimates for dyadic $b$-input $T(b)$ theorems

Salomone, Stephanie
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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The following deals with the $T(b)$ theorems of David, Journ\'e, and Semmes \cite{DJS} considered in a dyadic setting. We find sharp growth estimates for a global and a local dyadic $T(b)$ Theorem. We use multiscale analysis and Haar wavelets in the local case.; Comment: 15 pages

## ‣ Generalized sampling reconstruction from Fourier measurements using compactly supported shearlets

Ma, Jackie
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.703901%
In this paper we study the general reconstruction of a compactly supported function from its Fourier coefficients using compactly supported shearlet systems. We assume that only finitely many Fourier samples of the function are accessible and based on this finite collection of measurements an approximation is sought in a finite dimensional shearlet reconstruction space. We analyse this sampling and reconstruction process by a recently introduced method called generalized sampling. In particular by studying the stable sampling rate of generalized sampling we then show stable recovery of the signal is possible using an almost linear rate. Furthermore, we compare the result to the previously obtained rates for wavelets.

## ‣ Deep Convolutional Neural Networks Based on Semi-Discrete Frames

Wiatowski, Thomas; Bölcskei, Helmut
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Deep convolutional neural networks have led to breakthrough results in practical feature extraction applications. The mathematical analysis of these networks was pioneered by Mallat, 2012. Specifically, Mallat considered so-called scattering networks based on identical semi-discrete wavelet frames in each network layer, and proved translation-invariance as well as deformation stability of the resulting feature extractor. The purpose of this paper is to develop Mallat's theory further by allowing for different and, most importantly, general semi-discrete frames (such as, e.g., Gabor frames, wavelets, curvelets, shearlets, ridgelets) in distinct network layers. This allows to extract wider classes of features than point singularities resolved by the wavelet transform. Our generalized feature extractor is proven to be translation-invariant, and we develop deformation stability results for a larger class of deformations than those considered by Mallat. For Mallat's wavelet-based feature extractor, we get rid of a number of technical conditions. The mathematical engine behind our results is continuous frame theory, which allows us to completely detach the invariance and deformation stability proofs from the particular algebraic structure of the underlying frames.; Comment: Proc. of IEEE International Symposium on Information Theory (ISIT)...

## ‣ On convergence of general wavelet decompositions of nonstationary stochastic processes

Kozachenko, Yuriy; Olenko, Andriy; Polosmak, Olga
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The paper investigates uniform convergence of wavelet expansions of Gaussian random processes. The convergence is obtained under simple general conditions on processes and wavelets which can be easily verified. Applications of the developed technique are shown for several classes of stochastic processes. In particular, the main theorem is adjusted to the fractional Brownian motion case. New results on the rate of convergence of the wavelet expansions in the space $C([0,T])$ are also presented.; Comment: 24 pages, 2 figures, will appear in Electronic Journal of Probability

## ‣ A note on grid transfer operators for multigrid methods

Donatelli, Marco
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.703901%
The Local Fourier analysis (LFA) is a classic tool to prove convergence theorems for multigrid methods (MGMs). In particular, we are interested in optimality that is a convergence speed independent of the size of the involved matrices. For elliptic partial differential equations (PDEs), a well known optimality result requires that the sum of the orders of the grid transfer operators is not lower than the order of the PDE to solve. Analogously, when dealing with MGMs for Toeplitz matrices in the literature an optimality condition on the position and on the order of the zeros of the symbols of the grid transfer operators has been found. In this work we show that in the case of elliptic PDEs with constant coefficients, the two different approaches lead to an equivalent condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA, which allows to deal not only with differential problems but also for instance with integral problems. The equivalence of the two approaches gives the possibility of using grid transfer operators with different orders also for MGMs for Toeplitz matrices. We give also a class of grid transfer operators related to the B-spline's refinement equation and we study their geometric properties. This analysis suggests further links between wavelets and multigrid methods. A numerical experimentation confirms the correctness of the proposed analysis.

## ‣ Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets

Mourou, Mohamed Ali
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.703901%
We prove a Calder\'on reproducing formula for the Dunkl continuous wavelet transform on $\mathbb{R}$. We apply this result to derive new inversion formulas for the dual Dunkl-Sonine integral transform.

## ‣ A dynamic hybrid model based on wavelets and fuzzy regression for time series estimation

Zaafrane, Olfa; Mabrouk, Anouar Ben
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.703901%
In the present paper, a fuzzy logic based method is combined with wavelet decomposition to develop a step-by-step dynamic hybrid model for the estimation of financial time series. Empirical tests on fuzzy regression, wavelet decomposition as well as the new hybrid model are conducted on the well known $SP500$ index financial time series. The empirical tests show an efficiency of the hybrid model.; Comment: 15 pages, 15 figures, 2 tables

## ‣ Asymptotic normality of wavelet estimators of the memory parameter for linear processes

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.703901%
We consider linear processes, not necessarily Gaussian, with long, short or negative memory. The memory parameter is estimated semi-parametrically using wavelets from a sample $X_1,...,X_n$ of the process. We treat both the log-regression wavelet estimator and the wavelet Whittle estimator. We show that these estimators are asymptotically normal as the sample size $n\to\infty$ and we obtain an explicit expression for the limit variance. These results are derived from a general result on the asymptotic normality of the empirical scalogram for linear processes, conveniently centered and normalized. The scalogram is an array of quadratic forms of the observed sample, computed from the wavelet coefficients of this sample. In contrast with quadratic forms computed on the Fourier coefficients such as the periodogram, the scalogram involves correlations which do not vanish as the sample size $n\to\infty$.

## ‣ Central Limit Theorems for arrays of decimated linear processes

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.703901%
Linear processes are defined as a discrete-time convolution between a kernel and an infinite sequence of i.i.d. random variables. We modify this convolution by introducing decimation, that is, by stretching time accordingly. We then establish central limit theorems for arrays of squares of such decimated processes. These theorems are used to obtain the asymptotic behavior of estimators of the spectral density at specific frequencies. Another application, treated elsewhere, concerns the estimation of the long-memory parameter in time-series, using wavelets.

## ‣ Triebel-Lizorkin Spaces and Shearlets on the Cone in $\mathbb{R}^2$

Vera, Daniel
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.703901%
The shearlets are a special case of the wavelets with composite dilation that, among other things, have a basis-like structure and multi resolution analysis properties. These relatively new representation systems have encountered wide range of applications, generally surpassing the performance of their ancestors due to their directional sensitivity. However, little is known about their relation with spaces other than $L^2$. Here, we find a characterization of a kind of anisotropic inhomogeneous Triebel-Lizorkin spaces (to be defined) with the so called "shearlets on the cone" coefficients. We first prove the boundedness of the analysis and synthesis operators with the "traditional" shearlets coefficients. Then, with the development of the smooth Parseval frames of shearlets of Guo and Labate we are able to prove a reproducing identity, which was previously possible only for the $L^2$ case. We also find some embeddings of the (classical) dyadic spaces into these highly anisotropic spaces, and viceversa, for certain ranges of parameters. In order to keep a concise document we develop our results in the "weightless" case ($w=1$) and give hints on how to develop the weighted case.; Comment: 30 pages

## ‣ Bandlimited Spaces on Some 2-step Nilpotent Lie Groups With One Parseval Frame Generator

Oussa, Vignon
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.703901%
Let $N$ be a step two connected and simply connected non commutative nilpotent Lie group which is square-integrable modulo the center. Let $Z$ be the center of $N$. Assume that $N=P\rtimes M$ such that $P$, and $M$ are simply connected, connected abelian Lie groups, $M$ acts non-trivially on $P$ by automorphisms and $\dim P/Z=\dim M$. We study band-limited subspaces of $L^2(N)$ which admit Parseval frames generated by discrete translates of a single function. We also find characteristics of band-limited subspaces of $L^2(N)$ which do not admit a single Parseval frame. We also provide some conditions under which continuous wavelets transforms related to the left regular representation admit discretization, by some discrete set $\Gamma\subset N$. Finally, we show some explicit examples in the last section.

## ‣ Locally adaptive estimation of evolutionary wavelet spectra

Van Bellegem, Sébastien; von Sachs, Rainer
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.703901%
We introduce a wavelet-based model of local stationarity. This model enlarges the class of locally stationary wavelet processes and contains processes whose spectral density function may change very suddenly in time. A notion of time-varying wavelet spectrum is uniquely defined as a wavelet-type transform of the autocovariance function with respect to so-called autocorrelation wavelets. This leads to a natural representation of the autocovariance which is localized on scales. We propose a pointwise adaptive estimator of the time-varying spectrum. The behavior of the estimator studied in homogeneous and inhomogeneous regions of the wavelet spectrum.; Comment: Published in at http://dx.doi.org/10.1214/07-AOS524 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

## ‣ Framings and dilations

Larson, David R.; Szafraniec, Franciszek Hugon
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.703901%
The notion of framings, recently emerging in P. G. Casazza, D. Han, and D. R. Larson, Frames for Banach spaces, in {\em The functional and harmonic analysis of wavelets and frames} (San Antonio, TX, 1999), {\em Contemp. Math}. {\bf 247} (1999), 149-182 as generalization of the reconstraction formula generated by pairs of dual frames, is in this note extended substantially. This calls on refining the basic dilation results which still being in the flavor of {\em th\'eor\`eme principal} of B. Sz-Nagy go much beyond it.; Comment: The final version will appear in Acta Sci. Math (Szeged)

## ‣ An exact tree projection algorithm for wavelets

Cartis, Coralia; Thompson, Andrew
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
26.703901%
We propose a dynamic programming algorithm for projection onto wavelet tree structures. In contrast to other recently proposed algorithms which only give approximate tree projections for a given sparsity, our algorithm is guaranteed to calculate the projection exactly. We also prove that our algorithm has O(Nk) complexity, where N is the signal dimension and k is the sparsity of the tree approximation.; Comment: 4 pages, 1 figure

## ‣ On an identity by Chaundy and Bullard. I

Koornwinder, Tom H.; Schlosser, Michael J.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
26.703901%
An identity by Chaundy and Bullard writes 1/(1-x)^n (n=1,2,...) as a sum of two truncated binomial series. This identity was rediscovered many times. Notably, a special case was rediscovered by I. Daubechies, while she was setting up the theory of wavelets of compact support. We discuss or survey many different proofs of the identity, and also its relationship with Gauss hypergeometric series. We also consider the extension to complex values of the two parameters which occur as summation bounds. The paper concludes with a discussion of a multivariable analogue of the identity, which was first given by Damjanovic, Klamkin and Ruehr. We give the relationship with Lauricella hypergeometric functions and corresponding PDE's. The paper ends with a new proof of the multivariable case by splitting up Dirichlet's multivariable beta integral.; Comment: 20 pages; added in v3: more references to earlier occurrences of the identity and its multivariable analogue, combinatorial proof of the identity and extension to noninteger m,n, proof of multivariable identity by splitting up Dirichlet's multivariable beta integral

## ‣ Model selection for quantum homodyne tomography

Kahn, Jonas
Tipo: Artigo de Revista Científica