Wavelet expansions forBMOρ (w)‐functions

We give necessary and sufficient conditions on the wavelet coefficients of a function for being a member of some BMO_φ (w) space. We achieve this characterization for a wide variety of wavelet systems. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Approximation with Spline Generated Framelets

We characterize the approximation spaces associated with the best n-term approximation in L_p(R) by elements from a tight wavelet frame associated with a spline scaling function. The approximation spaces are shown to be interpolation spaces between L_p and classical Besov spaces, and the result coincides with the result for nonlinear approximation with an orthonormal wavelet with the same smoothness as the spline scaling function. We also show that, under certain conditions, the Besov smoothness can be measured in terms of the sparsity of expansions in the wavelet frame, just like the nonredundant wavelet case. However, the characterization now holds even for wavelet frame systems that do not have the usually required number of vanishing moments, e.g., for systems built through the Unitary Extension Principle, which can have no more than one vanishing moment. Using these results, we describe a fast algorithm that takes as input any function and provides a near sparsest expansion of it in the framelet system as well as approximants that reach the optimal rate of nonlinear approximation. Together with the existence of a fast algorithm, the absence of the need for vanishing moments may have an important qualitative impact for applications to signal compression, as high vanishing moments usually introduce a Gibbs-type phenomenon (or ringing artifacts)in the approximants.     

p-Adic orthogonal wavelet bases

We describe all MRA-based p-adic compactly supported wavelet systems forming an orthogonal basis for L ²(ℚ _p ). The text was submitted by the authors in English.     

<Emphasis Type="Italic">p</Emphasis>-Adic nonorthogonal wavelet bases

A method for constructing MRA-based p-adic wavelet systems that form Riesz bases in L ²(ℚ _p ) is developed. The method is implemented for an infinite family of MRAs.     

2-Adic wavelet bases

Within the theory of multiresolution analysis, a method of constructing 2-adic wavelet systems that form Riesz bases in L ²(ℚ₂) is developed. A realization of this method for some infinite family of multiresolution analyses leading to nonorthogonal Riesz bases is presented.     

Wavelet packets in Sobolev space

Wavelet packets in Sobolev space H^s (?) are constructed and their orthogonal properties are derived. Using convolution transform theory, boundedness results for the wavelet packets are obtained in the B_{p, ?} (?) space. Examples of wavelet packets in Sobolev space are given.     

Wavelet Approximation of Periodic Functions

We investigate expansions of periodic functions with respect to wavelet bases. Direct and inverse theorems for wavelet approximation in C and L_p norms are proved. For the functions possessing local regularity we study the rate of pointwise convergence of wavelet Fourier series. We also define and investigate the “discreet wavelet Fourier transform” (DWFT) for periodic wavelets generated by a compactly supported scaling function. The DWFT has one important advantage for numerical problems compared with the corresponding wavelet Fourier coefficients: while fast computational algorithms for wavelet Fourier coefficients are recursive, DWFTs can be computed by explicit formulas without any recursion and the computation is fast enough.     

Construction of wavelet sets with certain self-similarity properties

A measurable set Q ⊂ R ⁿ is a wavelet set for an expansive matrix A if F ⁻¹ _(ΧQ) is an A-dilation wavelet. Dai, Larson, and Speegle [7] discovered the existence of wavelet sets in R _n associated with any real n ×n expansive matrix. In this work, we construct a class of compact wavelet sets which do not contain the origin and which are, up to a certain linear transformation, finite unions of integer translates of an integral selfaffine tile associated with the matrix B = A ^t. Some of these wavelet sets may have good potential for applications because of their tractable geometric shapes.     

Multi-window dilation-and-modulation frames on the half real line

Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators, respectively, and have been studied extensively. However, dilation-and-modulation systems cannot be derived from wavelet or Gabor systems. This study aims to investigate a class of dilation-and-modulation systems in the causal signal space L²(ℝ₊). L²(ℝ₊) can be identified as a subspace of L²(ℝ), which consists of all L²(ℝ)-functions supported on ℝ₊ but not closed under the Fourier transform. Therefore, the Fourier transform method does not work in L²(ℝ₊). Herein, we introduce the notion of Θ_a-transform in L²(ℝ₊) and characterize the dilation-and-modulation frames and dual frames in L²(ℝ₊) using the Θ_a-transform; and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for L²(ℝ₊). Furthermore, it has been proven that an arbitrary frame of this form is always nonredundant whenever the number of the generators is 1 and is always redundant whenever the number is greater than 1. Finally, some examples are provided to illustrate the generality of our results.

     

Restricted Nonlinear Approximation

We introduce a new form of nonlinear approximation called restricted approximation . It is a generalization of n -term wavelet approximation in which a weight function is used to control the terms in the wavelet expansion of the approximant. This form of approximation occurs in statistical estimation and in the characterization of interpolation spaces for certain pairs of L _p and Besov spaces. We characterize, both in terms of their wavelet coefficients and also in terms of their smoothness, the functions which are approximated with a specified rate by restricted approximation. We also show the relation of this form of approximation with certain types of thresholding of wavelet coefficients. March 31, 1998. Date accepted: January 28, 1999.     

Perturbations of Banach Frames and Atomic Decompositions

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional — analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley — Wiener basis stability criteria and the perturbation theorem el kato. We introduce new and weaker conditions which ensure the desired stability. We then prove quality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form atomic decompositions for L²_w(IR), but cannot form Hilbert frames but L²_w(IR) unless the weight is trivial.     

Dyadic Bivariate Wavelet Multipliers in L^2（R@2）

The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L ²(ℝ²). In this paper, we choose $2I_2 = \left( {{*{20}c} 2 & 0 \\ 0 & 2 \\ } \right)$2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right) as the dilation matrix and consider the 2I ₂-dilation multivariate wavelet Φ = {ψ ₁, ψ ₂, ψ ₃}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f ₁, f ₂, f ₃} a dyadic bivariate wavelet multiplier if Y₁ = { F ^{- 1} ( f₁ [^(y₁ )] ),F ^{- 1} ( f₂ [^(y₂ )] ),F ^{- 1} ( f₃ [^(y₃ )] ) }\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\} is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ ₁, ψ ₂, ψ ₃}, where [^(f)]\hat f and F ⁻¹ denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets.     

Multiwavelet Frames from Refinable Function Vectors

Starting from any two compactly supported d-refinable function vectors in (L ₂(R)) ^r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L ₂(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.     

Near Best Tree Approximation

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [4,6] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. This paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. We accomplish this in the case that the approximation error is measured in L ₂, or in the case p2, in the Besov spaces B _p ⁰(L _p ), which are close to (but not the same as) L _p . Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients.     

A Parameter Selection Method for Wavelet Shrinkage Denoising

Thresholding estimators in an orthonormal wavelet basis are well established tools for Gaussian noise removal. However, the universal threshold choice, suggested by Donoho and Johnstone, sometimes leads to over-smoothed approximations.For the denoising problem this paper uses the deterministic approach proposed by Chambolle et al., which handles it as a variational problem, whose solution can be formulated in terms of wavelet shrinkage. This allows us to use wavelet shrinkage successfully for more general denoising problems and to propose a new criterion for the choice of the shrinkage parameter, which we call H-curve criterion. It is based on the plot, for different parameter values, of the B ₁ ¹(L ₁)-norm of the computed solution versus the L ₂-norm of the residual, considered in logarithmic scale. Extensive numerical experimentation shows that this new choice of shrinkage parameter yields good results both for Gaussian and other kinds of noise.     

WAVELET ESTIMATION IN NONPARAMETRIC MODEL UNDER MARTINGALE DIFFERENCE ERRORS

§1　IntroductionConsider the following heteroscedastic regression model:Yi =g(xi) +σiei,　1≤i≤n,(1.1)whereσ2i=f(ui) ,(xi,ui) are nonrandom design points,0≤x0 ≤x1 ≤...≤xn=1and0≤u0≤u1 ≤...≤un=1,Yi are the response variables,ei are random errors,and f(·) andg(·) are unknown functions defined on closed interval[0 ,1] .It is well known thatregression model has many applications in practical problems,sothe model (1.1) and its special cases have been studied extensively. For instance,…     

Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames

In this paper we shall characterize Sobolev spaces of an arbitrary order of smoothness using nonstationary tight wavelet frames for L ₂(ℝ). In particular, we show that a Sobolev space of an arbitrary fixed order of smoothness can be characterized in terms of the weighted ℓ₂-norm of the analysis wavelet coefficient sequences using a fixed compactly supported nonstationary tight wavelet frame in L ₂(ℝ) derived from masks of pseudosplines in [15]. This implies that any compactly supported nonstationary tight wavelet frame of L ₂(ℝ) in [15] can be properly normalized into a pair of dual frames in the corresponding pair of dual Sobolev spaces of an arbitrary fixed order of smoothness. Research supported in part by NSERC Canada under Grant RGP 228051. Research supported in part by Grant R-146-000-060-112 at the National University of Singapore.     

A necessary and sufficient condition for the biorthogonality of {ρ 1,ρ 2}

In this paper, we present a necessary and sufficient condition for the biorthogonality of a class of special functionsρ ₁ andρ ₂. The functions are useful in the theory of biorthogonal wavelet.     

N–Term Approximation in Anisotropic Function Spaces

In L₂(0, 1)²) infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one–dimensional biorthogonal wavelet bases on the interval (0, 1). Most well–known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.     

Universal optimality of digital nets and lattice designs

This article considers universal optimality of digital nets and lattice designs in a regression model. Based on the equivalence theorem for matrix means and majorization theory,the necessary and sufficient conditions for lattice designs being φp-and universally optimal in trigonometric function and Chebyshev polynomial regression models are obtained. It is shown that digital nets are universally optimal for both complete and incomplete Walsh function regression models under some specified conditions,and are...