Characterization of compactly supported refinable splines whose shifts form a Riesz basis

Based on (J. Approx. Theory 86 (1996) 240), we prove that the integer shifts of a multivariate blockwise polynomial φ(x) which is compactly supported and m-refinable form a Riesz basis if and only if . Here , c≠0 is a constant, B(x|v₁,v₂,…,v_k) is a multivariate box spline and the matrix (v₁,v₂,…,v_k) is unimodular.     

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 the construction of a class of bidimensional nonseparable compactly supported wavelets

Chui and Wang discussed the construction of one-dimensional compactly supported wavelets under a general framework, and constructed one-dimensional compactly supported spline wavelets. In this paper, under a mild condition, the construction of -wavelets is obtained.

     

紧支集分布和可加细向量的整平移的线性相关性

In this paper, the global and local linear independence of any compactly supported distributions by using time domain spaces ,and of refinable vectors by invariant linear spaces are investigated.     

Wavelet decomposition of the space of discrete periodic splines

An orthogonal basis for the spaceS _r ^m of discrete periodic splines is constructed. The wavelet decomposition of the spaceS _r ^m form=2 ^t is obtained using this basis. We derive recurrence formulas for the transformation from the decomposition with respect to the orthogonal basis to the wavelet decomposition, as well as recurrence formulas for the inverse transformation. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 712–720, May, 2000.     

On stable refinable function vectors with arbitrary support

Refinable function vectors with arbitrary support are considered. In particular, necessary conditions for stability are given and a characterization of the symbol associated with a stable refinable function vector in terms of the transfer operator is provided: this is a generalization of Gundy’s theorem to the vector case. The proof adapts the tools provided in [S. Saliani, On stability and orthogonality of refinable functions, Appl. Comput. Harmon. Anal. 21 (2006) 254–261]. Though complications arise from noncommuting matrix products, the fundamental ideas are the same.     

Infinite convolution products and refinable distributions on Lie groups

Sufficient conditions for the convergence in distribution of an infinite convolution product of measures on a connected Lie group with respect to left invariant Haar measure are derived. These conditions are used to construct distributions that satisfy where is a refinement operator constructed from a measure and a dilation automorphism . The existence of implies is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset such that for any open set containing and for any distribution on with compact support, there exists an integer such that implies If is supported on an -invariant uniform subgroup then is related, by an intertwining operator, to a transition operator on Necessary and sufficient conditions for to converge to , and for the -translates of to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of to functions supported on

     

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…     

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

     

Extreme problems on classes of polynomials and splines

By the methods of the theory of extremum problems, existence theorems for functions with given sequence of extreme values are proved.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 494–502, April, 1998.The author wishes to express his sincere gratitude to Professor V. M. Tikhomirov for setting the problem and for his permanent attention and help and also to the referee for the remarks, which aided in the amelioration of the present paper.     

Relations between the support of a compactly supported function and the exponential-polynomials spanned by its integer translates

The interrelation between the shape of the support of a compactly supported function and the space of all exponential-polynomials spanned by its integer translates is examined. The results obtained are in terms of the behavior of these exponential-polynomials on certain finite subsets ofZ ^s , which are determined by the support of the given function. Several applications are discussed. Among these is the construction of quasi-interpolants of minimal support and the construction of a piecewise-polynomial whose integer translates span a polynomial space which is not scale-invariant. As to polynomial box splines, it is proved here that in many cases a polynomial box spline admits a certain optimality condition concerning the space of the total degree polynomials spanned by its integer translates: This space is maximal compared with the spaces corresponding to other functions with the same supportCommunicated by Klaus Höllig.     

Regularity of refinable functions with exponentially decaying masks

The regularity of refinable functions is an important issue in all multiresolution analysis and has a strong impact on applications of wavelets to image processing, geometric and numerical solutions of elliptic partial differential equations. The purpose of this paper is to characterize the regularity of refinable functions with exponentially decaying masks and a dilation matrix whose eigenvalues have the same modulus. The main results of this paper are really extensions of some results in Cohen et al. (1999) [5], Jia (1999) [17] and Lorentz and Oswald (2000) [28].     

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.     

Univariate splines: Equivalence of moduli of smoothness and applications

Several results on equivalence of moduli of smoothness of univariate splines are obtained. For example, it is shown that, for any , , and , the inequality , , is satisfied, where is a piecewise polynomial of degree on a quasi-uniform (i.e., the ratio of lengths of the largest and the smallest intervals is bounded by a constant) partition of an interval. Similar results for Chebyshev partitions and weighted Ditzian-Totik moduli of smoothness are also obtained. These results yield simple new constructions and allow considerable simplification of various known proofs in the area of constrained approximation by polynomials and splines.

     

Non-self-adjoint sturm-liouville operators with matrix potentials

We obtain asymptotic formulas for non-self-adjoint operators generated by the Sturm-Liouville system and quasiperiodic boundary conditions. Using these asymptotic formulas, we obtain conditions on the potential for which the system of root vectors of the operator under consideration forms a Riesz basis.     

Characterization of rationally sampled quaternionic dual Gabor frames

This paper addresses quaternionic dual Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. For a general overcomplete quaternionic Gabor frame with the product of time-frequency shift parameters not equal to $\frac{1}{2}$, we show that its corresponding frame and translation operators do not commute, which leads to its canonical dual frame not having the Gabor structure, but it may have other dual frames with Gabor structure. We characterize when two quaternionic Gabor Bessel sequences form a pair of dual frames, and present a class of quaternionic dual Gabor frames. We also characterize quaternionic Gabor Riesz bases and prove that their canonical dual frames have Gabor structure.     

Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two-scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m+1 and defectr, first considered by Lipow and Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degreem and defectr. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution.     

A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling

The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established.     

多项式组的特征分解

任一多项式理想的特征对是指由该理想的约化字典序Grobner基G和含于其中的极小三角列C构成的有序对(G,C).当C为正则列或正规列时,分别称特征对(G,C)为正则的或正规的.当G生成的理想与C的饱和理想相同时,称特征对(G,C)为强的.一组多项式的(强)正则或(强)正规特征分解是指将该多项式组分解为有限多个(强)正则或(强)正规特征对,使其满足特定的零点与理想关系.本文简要回顾各种三角分解及相应零点与理想分解的理论和方法,然后重点介绍(强)正则与(强)正规特征对和特征分解的性质,说明三角列、Ritt特征列和字典序Grobner基之间的内在关联,建立特征对的正则化定理以及正则、正规特征对的强化方法,进而给出两种基于字典序Grobner基计算、按伪整除关系分裂和构建、商除可除理想等策略的(强)正规与(强)正则特征分解算法.这两种算法计算所得的强正规与强正则特征对和特征分解都具有良好的性质,且能为输入多元多项式组的零点提供两种不同的表示.本文还给出示例和部分实验结果,用以说明特征分解方法及其实用性和有效性.