Compactly supported (bi)orthogonal wavelets generated by interpolatory refinable functions

This paper provides several constructions of compactly supported wavelets generated by interpolatory refinable functions. It was shown in [7] that there is no real compactly supported orthonormal symmetric dyadic refinable function, except the trivial case; and also shown in [10,18] that there is no compactly supported interpolatory orthonormal dyadic refinable function. Hence, for the dyadic dilation case, compactly supported wavelets generated by interpolatory refinable functions have to be biorthogonal wavelets. The key step to construct the biorthogonal wavelets is to construct a compactly supported dual function for a given interpolatory refinable function. We provide two explicit iterative constructions of such dual functions with desired regularity. When the dilation factors are larger than 3, we provide several examples of compactly supported interpolatory orthonormal symmetric refinable functions from a general method. This leads to several examples of orthogonal symmetric (anti‐symmetric) wavelets generated by interpolatory refinable functions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Symmetric orthonormal scaling functions and wavelets with dilation factor 4

It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d=4. Several examples of such orthonormal scaling functions are provided in this paper. In particular, two examples of C ¹ orthonormal scaling functions, which are symmetric about 0 and 1/6, respectively, are presented. We will then discuss how to construct symmetric wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

[0,1]区间上的r重正交多小波基

本文利用L2（R）上的紧支撑正交的多尺度函数和多小波构造出有限区间[0,1]上的正交多尺度函数及相应的正交多小波.本文构造的逼近空间Vj[0,1]与相应的小波子空间Wj[0,1]具有维数相同的特点,从而给它的应用带来巨大方便.最后给出重数为2时的[0,1]区间上的正交多小波基构造算例.     

Symmetric–Antisymmetric Orthonormal Multiwavelets and Related Scalar Wavelets

For compactly supported symmetric–antisymmetric orthonormal multiwavelet systems with multiplicity 2, we first show that any length-2Nmultiwavelet system can be constructed from a length-(2N+1) multiwavelet system and vice versa. Then we present two explicit formulations for the construction of multiwavelet functions directly from their associated multiscaling functions. This is followed by the relationship between these multiscaling functions and the scaling functions of related orthonormal scalar wavelets. Finally, we present two methods for constructing families of symmetric–antisymmetric orthonormal multiwavelet systems via the construction of the related scalar wavelets.     

A note on orthonormal polynomial bases and wavelets

A simple and explicit construction of an orthnormal trigonometric polynomial basis in the spaceC of continuous periodic functions is presented. It consists simply of periodizing a well-known wavelet on the real line which is orthonormal and has compactly supported Fourier transform. Trigonometric polynomials resulting from this approach have optimal order of growth of their degrees if their indices are powers of 2. Also, Fourier sums with respect to this polynomial basis are projectors onto subspaces of trigonometric polynomials of high degree, which implies almost best approximation properties.     

Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function φ which has sufficient regularity and which is fundamental for interpolation [that means, φ(0)=1 and φ(α)=0 for all α∈ Z ^s ∖{0}].
Low regularity examples of such functions have been obtained numerically by several authors, and a more general numerical scheme was given in [1]. This article presents several schemes to construct compactly supported fundamental refinable functions, which have higher regularity, directly from a given, continuous, compactly supported, refinable fundamental function φ. Asymptotic regularity analyses of the functions generated by the constructions are given.The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces.
A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in constructing biorthogonal wavelets. Combined with the biorthogonal wavelet construction algorithm for a pair of dual refinable functions given in [2], we are able to obtain symmetrical compactly supported multivariate biorthogonal wavelets which have arbitrarily high regularity. Several examples are computed.     

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.     

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.     

A construction of convex functions

We describe a construction of convex functions on infinite-dimensional spaces and apply this construction to give an illustration to a theorem of Borwein-Fabian from [1]. Namely, we give a simple explicit example of a continuous convex function on l _p , p ≥ 1, which is everywhere compactly differentiable, but not Fréchet differentiable at zero.     

A family of orthonormal wavelet bases with dilation factor 4

In this paper, we study a method for the construction of orthonormal wavelet bases with dilation factor 4. More precisely, for any integer M>0, we construct an orthonormal scaling filter mM(ξ) that generates a mother scaling function ?M, associated with the dilation factor 4. The computation of the different coefficients of ²|mM(ξ)| is done by the use of a simple iterative method. Also, this work shows how this construction method provides us with a whole family of compactly supported orthonormal wavelet bases with arbitrary high regularity. A first estimate of α(M), the asymptotic regularity of ?M is given by α(M)∼0.25M. Examples are provided to illustrate the results of this work.     

Compactly supported multiwindow dual Gabor frames of rational sampling density

We consider multiwindow Gabor systems (G _N ; a, b) with N compactly supported windows and rational sampling density N/ab. We give another set of necessary and sufficient conditions for two multiwindow Gabor systems to form a pair of dual frames in addition to the Zibulski–Zeevi and Janssen conditions. Our conditions come from the back transform of Zibulski–Zeevi condition to the time domain but are more informative to construct window functions. For example, the masks satisfying unitary extension principle (UEP) condition generate a tight Gabor system when restricted on [0, 2] with a?=?1 and b?=?1. As another application, we show that a multiwindow Gabor system (G _N ; 1, 1) forms an orthonormal basis if and only if it has only one window (N?=?1) which is a sum of characteristic functions whose supports ‘essentially’ form a Lebesgue measurable partition of the unit interval. Our criteria also provide a rich family of multiwindow dual Gabor frames and multiwindow tight Gabor frames for the particular choices of lattice parameters, number and support of the windows. (Section 4)     

紧支撑正交对称和反对称小波的构造

１．引言 近年来,人们分别从数学和信号的观点对正交小波进行了广泛的研究．尤其是２尺度小波,它克服了短时 Ｆｏｕｒｉｅｒ变换的一些缺陷．目前最常用的 ２尺度小波是 Ｄａｕｂｅｃｈｉｅｓ 小波,但 ２尺度小波也存在一些问题：如 Ｄａｕｂｅｃｈｉｅｓ［２］已证明了除 Ｈａａｒ小波外不存在既正交又对称的紧支撑 ２尺度小波．因此人们提出了 ａ尺度小波理论［３］－［６］,文献［４］－［６］对 ４尺度小波迸行研究．本文的目的是研究４尺度因子时紧支撑正交对称和反对称小波的构造方法．并指出对同一紧支撑正交对称尺度函数而言,…     

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.     

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.     

Pairs of Dual Wavelet Frames from Any Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R) with dilation factor d,we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L₂(R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L₂(R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates (d · – k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments.We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.     

Hardy-type inequalities on planar and spatial open sets

Several Hardy-type inequalities with explicit constants are proved for compactly supported smooth functions on open sets in the Euclidean space ? ⁿ .     

A nonlocal nonlinear diffusion equation in higher space dimensions

We study the initial-value problem for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation, in the whole RN, N?1, or in a bounded smooth domain with Neumann or Dirichlet boundary conditions. First, we prove the existence, uniqueness and the validity of a comparison principle for solutions of these problems. In RN we show that if initial data is bounded and compactly supported, then the solutions is compactly supported for all positive time t, this implies the existence of a free boundary. Concerning the Neumann problem, we prove that the asymptotic behavior of the solutions as t→∞, they converge to the mean value of the initial data. For the Dirichlet problem we prove that the asymptotic behavior of the solutions as t→∞, they converge to zero.     

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.     

A CHARACTERIZATION OF N-DIMENSIONAL DAUBECHIES TYPE TENSOR PRODUCT WAVELET

1. IntroductionSince the iniroduction by Daubechiesl1l of compactly supported orthogonal waveletbases in R with arbitrary high smoothness, various new wavelet bases have been constructedand aPplied successfully in image processing, numerical computation, statistics, etc. Manyof these applications, such as image compression, empIoy wavelet bases in R2. Virtuallyall of these bases are separable, that is, the bivariate functions are simply tensor productsof univariate basis functions. A separab…     

A COMPACTLY SUPPORTED ORTHONORMAL WAVELET OF NON-TENSOR TYPE

Abstract. In this note we construct an example of compactly supported orthonormal wavelets of non-tensor type from a multiresolutlon of