Bulk universality for Wigner matrices

We consider N × N Hermitian Wigner random matrices H where the probability density for each matrix element is given by the density ν(x) = e^?U(x). We prove that the eigenvalue statistics in the bulk are given by the Dyson sine kernel provided that U ∈ C⁶( input amssym $Bbb R$ ) with at most polynomially growing derivatives and ν(x) ≥ Ce^?C|x| for x large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales. © 2010 Wiley Periodicals, Inc.     

Deconvolving compactly supported densities

This paper addresses the statistical problem of density deconvolution under the condition that the density to be estimated has compact support. We introduce a new estimation procedure, which establishes faster rates of convergence for smooth densities as compared to the optimal rates for smooth densities with unbounded support. This framework also allows us to relax the usual condition of known error density with non-vanishing Fourier transform, so that a nonparametric class of densities is valid; therefore, even the shape of the noise density need not be assumed. These results can also be generalized for fast decaying densities with unbounded support. We prove optimality of the rates in the underlying experiment and study the practical performance of our estimator by numerical simulations.      

Bulk universality for generalized Wigner matrices

Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure ν _ij with a subexponential decay. Let ${\sigma_{ij}^2}$ be the variance for the probability measure ν _ij with the normalization property that ${\sum_{i} \sigma^2_{ij} = 1}$ for all j. Under essentially the only condition that ${c\le N \sigma_{ij}^2 \le c^{-1}}$ for some constant c?>?0, we prove that, in the limit N → ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale M ^?1.     

Semiclassical resolvent bounds for compactly supported radial potentials

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator $?h^2\mathrm{\Delta }+V(|x|)?E$ in dimension $n\ge 2$, where $h,E>0$, and $V:[0,\infty )\to \mathbb{R}$ is $L^{\infty }$ and compactly supported. The weighted resolvent norm grows no faster than $\exp ?(C{h}^{?1})$, while an exterior weighted norm grows $～h^{?1}$. We introduce a new method based on the Mellin transform to handle the two-dimensional case.     

Gibbs' phenomenon for nonnegative compactly supported scaling vectors

This paper considers Gibbs' phenomenon for scaling vectors in L²(R). We first show that a wide class of multiresolution analyses suffer from Gibbs' phenomenon. To deal with this problem, in [Contemp. Math. 216 (1998) 63-79], Walter and Shen use an Abel summation technique to construct a positive scaling function Pr, 0<r<1, from an orthonormal scaling function ? that generates V₀. A reproducing kernel can in turn be constructed using Pr. This kernel is also positive, has unit integral, and approximations utilizing it display no Gibbs' phenomenon. These results were extended to scaling vectors and multiwavelets in [Proceedings of Wavelet Analysis and Multiresolution Methods, 2000, pp. 317-339]. In both cases, orthogonality and compact support were lost in the construction process. In this paper we modify the approach given in [Proceedings of Wavelet Analysis and Multiresolution Methods, 2000, pp. 317-339] to construct compactly supported positive scaling vectors. While the mapping into V₀ associated with this new positive scaling vector is not a projection, the scaling vector does produce a Riesz basis for V₀ and we conclude the paper by illustrating that expansions of functions via positive scaling vectors exhibit no Gibbs' phenomenon.     

Multivariate compactly supported biorthogonal spline wavelets

We study biorthogonal bases of compactly supported wavelets constructed from box splines in ℝ ^N with any integer dilation factor. For a suitable class of box splines we write explicitly dual low-pass filters of arbitrarily high regularity and indicate how to construct the corresponding high-pass filters (primal and dual). Received: August 23, 2000; in final form: March 10, 2001?Published online: May 29, 2002     

Characterization of compactly supported refinable splines

We prove that a compactly supported spline functionφ of degree k satisfies the scaling equation $ \phi (x) = \sum _{n = 0}^N c(n)\phi (mx - n) $ for some integerm ≥ 2, if and only if $ \phi (x) = \sum _n p(n)B_k (x - n) $ wherep(n) are the coefficients of a polynomialP(z) such that the roots ofP(z)(z - 1)^k+1 TM are mapped into themselves by the mappingz →z ^m, andB _k is the uniform B-spline of degreek. Furthermore, the shifts ofφ form a Riesz basis if and only ifP is a monomial.     

On infinitely smooth compactly supported almost-wavelets

Translated from Matematicheskie Zametki, Vol. 56, No. 3, pp. 3–12, September, 1994.     

Biorthogonal bases of compactly supported wavelets

Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.     

Orthonormal bases of compactly supported wavelets

We construct orthonormal bases of compactly supported wavelets, with arbitrarily high regularity. The order of regularity increases linearly with the support width. We start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction. The construction then follows from a synthesis of these different approaches.     

紧支半正交小波的构造

§ 1　IntroductionIt is well known that wavelets with dilation factor two can be constructed from amultiresolution analysis and lots of their applications have been found.It is also knownthat wavelets with general dilation factor M may be constructed from the multiresolutionanalysis with dilation factor M≥ 2 [1— 3] .Wavelets are closely related to M-channel filterbanks[4,5] ,so some important applications such as in audio coding and communication aredeveloped.Semi-orthogonal wavelets const…     

Classification of edges using compactly supported shearlets

We analyze the detection and classification of singularities of functions $f={\chi }_S$, where $S?{\mathbb{R}}^d$ and $d=2,3$. It will be shown how the set ?S can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if ?S is a $d?1$ dimensional piecewise smooth manifold with $d=2$ or 3, we will classify smooth and non-smooth components of ?S. This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in 3 dimensions by compactly supported shearlets. Finally, geometric properties of ?S such as curvature are described in terms of the continuous shearlet transform of f.     

Construction of multivariate compactly supported orthonormal wavelets

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I_2×2 in the bivariate setting and show how to construct compactly supported functions ψ₁,. . .,ψ_n with n>3 such that {2^kψ_j(2^kx−ℓ,2^ky−m), k,ℓ,m∈Z, j=1,. . .,n} is an orthonormal basis for L₂(ℝ²). Here, n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C15, 42C30.     

Craya decomposition using compactly supported biorthogonal wavelets

We present a new local Craya–Herring decomposition of three-dimensional vector fields using compactly supported biorthogonal wavelets. Therewith vector-valued function spaces are split into two orthogonal components, i.e., curl-free and divergence-free spaces. The latter is further decomposed into toroidal and poloidal parts to decorrelate horizontal from vertical contributions which are of particular interest in geophysical turbulence. Applications are shown for isotropic, rotating and stratified turbulent flows. A comparison between isotropic and anisotropic orthogonal Craya–Herring wavelets, built in Fourier space and thus not compactly supported, is also given.     

Construction of trivariate biorthogonal compactly supported wavelets

In this paper, first we introduce trivariate multiresolution analysis and trivariate biorthogonal wavelets. A sufficient condition on the existence of a pair of trivariate biorthogonal scaling functions is derived. Then, the pair of nonseparable or separable trivariate biorthogonal wavelets can be achieved from the pair of trivariate biorthogonal scaling functions.     

Tight frames of compactly supported multivariate multi-wavelets

This paper is devoted to the study and construction of compactly supported tight frames of multivariate multi-wavelets. In particular, a necessary condition for their existence is derived to provide some useful guide for constructing such MRA tight frames, by reducing the factorization task of the associated polyphase matrix-valued Laurent polynomial to that of certain scalar-valued non-negative ones. We illustrate our construction method with examples of both multivariate scalar- and vector-valued subdivision schemes. Since our constructions for C¹ and C² piecewise cubic schemes are quite involved, we also include the corresponding Matlab code in the Appendix.     

Geodesics in the compactly supported Hamiltonian diffeomorphism group

Partially supported by YTF of Edu. Comm. and NSF of China     

Design of compactly supported trivariate orthogonal wavelets

In this paper, a method is developed for constructing compactly supported trivariate orthogonal wavelets from univariate orthogonal wavelets, essential idea of the approach is permutation of conjugate quadrature filter. Nonseparable and separable wavelets can be achieved from univariate orthogonal wavelets. Two examples are given to demonstrate this method.     

Hyperbolic sigma-pi neural network operators for compactly supported continuous functions

It is the aim of this contribution to continue our investigations on a special family of hyperbolic-type linear operators (here, for compactly supported continuous functions on IR ⁿ ) which immediately can be interpreted as concrete real-time realizations of three-layer feedforward neural networks with sigma-pi units in the hidden layer. To indicate how these results are connected with density results we start with some introductory theorems on this topic. Moreover, we take a detailed look at the complexity of the generated neural networks in order to achieve global -accuracy.     

A characterization of compactly supported orthonormal wavelets

In order to construct a compactly supported wavelet family inL ₂ with the help of the so-called (N stage) dilation equation, one has to impose some well-known constraints on its coefficients. To make the family an orthonormal basis ofL ₂, an additional condition has to be fulfilled. In this paper the condition has been rigorously simplified and made explicit. Examples are given for low dimensionsN.