The Method of Virtual Components in the Multivariate Setting

We describe the so-called method of virtual components for tight wavelet framelets to increase their approximation order and vanishing moments in the multivariate setting. Two examples of the virtual components for tight wavelet frames based on bivariate box splines on three or four direction mesh are given. As a byproduct, a new construction of tight wavelet frames based on box splines under the quincunx dilation matrix is presented.     

On Dual Wavelet Tight Frames

A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton's result on wavelet tight frames inL²( ) is generalized to then-dimensional case. Two ways of constructing certain dual wavelet tight frames inL²( ⁿ) are suggested. Finally, examples of smooth wavelet tight frames inL²( ) andH²( ) are provided. In particular, an example is given to demonstrate that there is a function ψ whose Fourier transform is positive, compactly supported, and infinitely differentiable which generates a non-MRA wavelet tight frame inH²( ).     

Construction of two-direction tight wavelet frames

We investigate the construction of two-direction tight wavelet frames First, a sufficient condition for a two-direction refinable function generating two-direction tight wavelet frames is derived. Second, a simple constructive method of two-direction tight wavelet frames is given. Third, based on the obtained two-direction tight wavelet frames, one can construct a symmetric multiwavelet frame easily. Finally, some examples are given to illustrate the results.     

Wavelet frames and shift-invariant subspaces of periodic functions

A general approach based on polyphase splines, with analysis in the frequency domain, is developed for studying wavelet frames of periodic functions of one or higher dimensions. Characterizations of frames for shift-invariant subspaces of periodic functions and results on the structure of these subspaces are obtained. Starting from any multiresolution analysis, a constructive proof is provided for the existence of a normalized tight wavelet frame. The construction gives the minimum number of wavelets required. As an illustration of the approach developed, the one-dimensional dyadic case is further discussed in detail, concluding with a concrete example of trigonometric polynomial wavelet frames.     

Tight compactly supported wavelet frames of arbitrarily high smoothness

Based on Ron and Shen's new method for constructing tight wave-let frames, we show that one can construct, for any dilation matrix, and in any spatial dimension, tight wavelet frames generated by compactly supported functions with arbitrarily high smoothness.

     

Compactly supported tight affine spline frames in

The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of ``mother wavelets', however, increases with the increase of the required smoothness.

Two bivariate constructions, of potential practical value, are highlighted. In both, the wavelets are derived from four-direction mesh box splines that are refinable with respect to the dilation matrix .

     

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.     

On fundamental solutions for multivariate singular interpolation

The objective of this paper is to give a constructive approach to the solution of the fundamental functions for cardinal interpolation from a shift-invariant space generated by the (multi-)integer translates of some compactly supported function whose polynomial symbol has a non-empty zero set. This problem was first introduced by Chui, Diamond, and Raphael, where explicit solutions were given for various zero sets. Later, de Boor, Höllig, and Riemenschneider gave an existence proof for zero sets which are more general. In this paper, we give an integral representation of the fundamental solutions that can be made explicit in some cases and we will also give a growth condition of such fundamental solutions. The four-directional box splines will be used as an illustrative example.     

An Algebraic Perspective on Multivariate Tight Wavelet Frames

Recent advances in real algebraic geometry and in the theory of polynomial optimization are applied to answer some open questions in the theory of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) are given in terms of Hermitian sums of squares of certain nonnegative Laurent polynomials and in terms of semidefinite programming. These formulations merge recent advances in real algebraic geometry and wavelet frame theory and lead to an affirmative answer to the long-standing open question of the existence of tight wavelet frames in dimension d=2. They also provide, for every d, efficient numerical methods for checking the existence of tight wavelet frames and for their construction. A class of counterexamples in dimension d=3 show that, in general, the so-called sub-QMF condition is not sufficient for the existence of tight wavelet frames. Stronger sufficient conditions for determining the existence of tight wavelet frames in dimension d≥3 are derived. The results are illustrated on several examples.     

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.     

Affine Frames of Multivariate Box Splines and their Affine Duals

We give a simple and explicit construction of primal and dual wavelet filters based on refinable multivariate splines (with respect to dilation matrices M) such that the corresponding wavelet functions generate dual affine frames of arbitrarily high regularity. Furthermore, the number of wavelets does not depend on the regularity. We apply the method also to generalized B-splines.     

向量值双正交小波的存在性及滤波器的构造

引进了向量值多分辨分析与向量值双正交小波的概念.讨论了向量值双正交小波的存在性.运用多分辨分析和矩阵理论,给出一类紧支撑向量值双正交小波滤波器的构造算法.最后,给出4-系数向量值双正交小波滤波器的的构造算例.     

A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution

Given a multivariate compactly supported distribution, we derive here a necessary and sufficient condition for the global linear independence of its integer translates. This condition is based on the location of the zeros of =the Fourier-Laplace transform of. The utility of the condition is demonstrated by several examples and applications, showing, in particular, that previous results on box splines and exponential box splines can be derived from this condition by a simple combinatorial argument.Communicated by Carl de Boor.     

Embedding and wavelet decomposition of spaces of minimal splines

On a sequence of embedded nonuniform grids, chains of embedded spaces of minimal splines (not necessarily polynomial) are constructed. The wavelet decomposition is given. The basis wavelets are compactly supported and admit simple analytic representation. The corresponding decomposition and reconstruction formulas are derived. The variety of spaces under consideration is identified with the variety of complete sequences of points of the direct product of an interval and a projective plane. Bibliography: 20 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 15–31     

N维2带小波紧框架的构造

给出了$m$个函数生成$N$维2带小波紧框架的充分条件和$N$维2带小波紧框架的显式构造算法, 讨论了小波紧框架的分解算法与重构算法. 提出的构造方法很有普遍性, 容易推广到$N(N\geq2)$维$M(M\geq 2)$带小波紧框架的情形,也可以得到类似的小波紧框架的分解算法与重构算法.     

Compactly Supported Tight Frames Associated with Refinable Functions

It is well known that in applied and computational mathematics, cardinal B-splines play an important role in geometric modeling (in computer-aided geometric design), statistical data representation (or modeling), solution of differential equations (in numerical analysis), and so forth. More recently, in the development of wavelet analysis, cardinal B-splines also serve as a canonical example of scaling functions that generate multiresolution analyses of L²(−∞,∞). However, although cardinal B-splines have compact support, their corresponding orthonormal wavelets (of Battle and Lemarie) have infinite duration. To preserve such properties as self-duality while requiring compact support, the notion of tight frames is probably the only replacement of that of orthonormal wavelets. In this paper, we study compactly supported tight frames Ψ={ψ¹,…,ψ^N} for L²(−∞,∞) that correspond to some refinable functions with compact support, give a precise existence criterion of Ψ in terms of an inequality condition on the Laurent polynomial symbols of the refinable functions, show that this condition is not always satisfied (implying the nonexistence of tight frames via the matrix extension approach), and give a constructive proof that when Ψ does exist, two functions with compact support are sufficient to constitute Ψ, while three guarantee symmetry/anti-symmetry, when the given refinable function is symmetric.     

Symmetric Framelets

We study tight wavelet frames associated with symmetric compactly supported refinable functions, which are obtained with the unitary extension principle . We give a criterion for the existence of two symmetric or antisymmetric compactly supported framelets. All refinable masks of length up to 6 satisfying this criterion are found.     

On Multivariate Compactly Supported Bi-Frames

In this article, we construct compactly supported multivariate pairs of dual wavelet frames, shortly called bi-frames, for an arbitrary dilation matrix. Our construction is based on the mixed oblique extension principle, and it provides bi-frames with few wavelets. In the examples, we obtain optimal bi-frames, i.e., primal and dual wavelets are constructed from a single fundamental refinable function, whose mask size is minimal w.r.t. sum rule order and smoothness. Moreover, the wavelets reach the maximal approximation orderw.r.t. the underlying refinable function. For special dilation matrices, we derive very simple but optimal arbitrarily smooth bi-frames in arbitrary dimensions with only two primal and dual wavelets.