正交多小波的构造

1 引言 在小波的构造和应用中,对于2尺度单一小波已有相当成熟的理论,特别是在小波构造方面,若知道正交单一尺度函数,相应的单一小波是很容易构造出的。对于a尺度紧支撑多小波,如何从已知的a尺度紧支撑多重尺度函数构造出相应的多小波,到目前为止尚没有一般的构造方法。W.Lanton等用仿酉矩阵扩充的方法构造出相应的多小     

二元正交小波的构造

高维小波是处理多维信息的工具。本文给出的构造紧支撑不可分二元正交小波函数的算法，当尺度函数和符号中所含因子[(1 z1/2)(1 z2/2)]^2的幂指数r越高时，尺度函数越光滑。     

对称反对称紧支撑正交多小波的构造

对于给定的对称反对称紧支撑正交r重尺度函数,给出一种构造对称反对称紧支撑正交多小波的方法.通过此方法构造的多小波与尺度函数有相同的对称性与反对称性,并且给出算例.     

紧支撑的最小能量框架

基于仿酉矩阵扩充方法,本文构造了一维d带紧支撑的最小能量框架,给出了一维d带紧支撑最小能量框架的显式构造算法.所构造的最小能量框架的支撑不超过尺度函数的支撑.当所给的尺度函数具有对称性时,研究了紧支撑对称最小能量框架的结构.最后,构造了两个算例.     

三元正交小波的构造

高维小波分析是分析和处理多维数字信号的有力工具.张量积小波有其自身的缺点.本文给出构造紧支撑三元不可分正交尺度函数和正交小波函数的新算法.当尺度函数的符号中含有因子1 z1221 z2221 z322的幂指数越高时,尺度函数越光滑.     

基于Cayley变换的紧支撑二元正交小波滤波器组的构造

构造正交滤波器组,在多相域里就等价于构造仿酉矩阵,而仿酉矩阵的构造涉及到非线性方程组的求解.通过对Cayley变换的研究,把仿酉矩阵的构造转换为更易构造的仿斜厄米特矩阵,基于这种变换构造了二元紧支撑正交小波滤波器组,并给出了算例.     

三进制双正交对称小波的设计

本文给出了一种三进制双正交对称小波的设计方法.在给定插值紧支撑对称尺度函数的情况下,指出了如果对偶尺度函数同为紧支撑插值的,则它们同为1-型对称.并且给出了对偶尺度函数为紧支撑插值和非插值情况下的通解计算公式.还提出了频率优化方法设计对偶尺度函数和小波函数,把双正交条件归结为线性约束的二次规划问题,最后通过线性方程组来求解.对于小波函数本文也给出了一组特解公式.     

紧支撑正交的二维小波

基于Householder矩阵扩充,构造了紧支撑正交的二维小波,所构造小波函数的支撑不超过尺度函数的支撑,并且给出了容易实施的显式构造算法．另外,还通过构造反例说明Riesz定理不适用于二元三角多项式．最后,构造了算例．     

矩阵的正交扩充与多正交小波

给出一种由a尺度紧支撑正交多尺度函数构造短支撑正交多小波的方法,其过程仅仅应用矩阵的正交扩充和求解方程组。如果r重尺度函数的支撑区间较大,可以将其转化为ar重短支撑情形,从而使得本文的方法适用于任意紧支撑正交多小波的构造,文后给出多小波的构造算例。     

仿酉对称矩阵的构造及对称正交多小波滤波带的参数化

仿酉矩阵在小波、多小波、框架的构造中发挥了重要的作用.本文给出仿酉对称矩阵(简记为p.s.m.)的显式构造算法,其中仿酉对称矩阵是元素为对称或反对称多项式的仿酉矩阵.基于已构造的p.s.m.和已知的正交对称多小波(简记为o.s.m.),给出o.s.m.的参数化.恰当地选择一些参数,可得到具有一些优良性质的o.s.m.,例如Armlet.最后作这一个算例,构造出一类对称的Chui-Lian Armlet滤波带.     

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L²(R, C^s) from the orthogonal vector-valued wavelet packets.     

具有奇异性的周期椭圆算子的绝对连续性

该文研究周期椭圆算子sun from(j,l=1) to d D_(jw)(x)a_(jl)D_l+V(x)在R~d(d≥3)中的谱性质,其中A=(a_(jl))是d×d阶的实常值正定矩阵,V(x)和w(x)是关于相同格点的周期标量函数,并且w(x)是正的.利用文中第一作者建立的d-环面上的一致Sobolev不等式,证明了该算子的谱是纯绝对连续的,如果V∈L_(loc)~(2pd/(d+2p))(R~d)且w∈A_(1+α)~(p,∞)(T~d)∩L~∞(T~d)(α0,p≥d),或者V∈L_(loc)~(2d/3)/(R~d),ω∈C~1(T~d),或者V∈L_(loc)~(d/2)(R~d),w∈L_(2,loc)~(d/2)(T~d).     

Heisenberg群上无穷远处的集中列紧原理和具有Sobolev临界指数的p - 次Laplace方程多解的存在性

通过建立Heisenberg群上无穷远处的集中列紧原理, 研究了如下$p$ -次Laplace方程 -Δ_{H, p}u=λg(ξ)|u|^q-2u+f (ξ)|u|^p*-2u,在Hⁿ上, u∈ D^{1, p}(Hⁿ), 其中ξ∈Hⁿ,λ∈R,1j, 且m, j为整数.     

一类带反应项的 Othmer-Stevens型趋化模型解的存在性

该文讨论了一类带反应项的Othmer-Stevens 型趋化模型的初边值问题 {∂u/∂t=D∨(u∨lnu/Φ(x, t, w))+ f(x, t, u), ∂w/∂t=g(x, t, u, w), u∨lnu/Φ(x, t, w) ?n^→=0. 证明了: 如果边界∂Ω ∈C^2+β, 函数Φ(x, t , w), f(x, t, u) 和 g(x, t, u, w)充分光滑,则该系统存在唯一解.     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

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.     

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.     

A study on orthogonal two-direction vector-valued wavelets and two-direction wavelet packets

In this paper, the notion of two-direction vector-valued multiresolution analysis and the two-direction orthogonal vector-valued wavelets are introduced. The definition for two-direction orthogonal vector-valued wavelet packets is proposed. An algorithm for constructing a class of two-direction orthogonal vector-valued compactly supported wavelets corresponding to the two-direction orthogonal vector-valued compactly supported scaling functions is proposed by virtue of matrix theory and time-frequency analysis method. The properties of the two-direction vector-valued wavelet packets are investigated. At last, the direct decomposition relation for space L₂(R)^r is presented.     

A class of compactly supported orthogonal symmetric complex wavelets with dilation factor 3

When approximation order is an odd positive integer a simple method is given to construct compactly supported orthogonal symmetric complex scaling function with dilation factor 3. Two corresponding orthogonal wavelets, one is symmetric and the other is antisymmetric about origin, are constructed explicitly. Additionally, when approximation order is an even integer 2, we also give a method to construct compactly supported orthogonal symmetric complex wavelets. In the end, there are several examples that illustrate the corresponding results.     

具临界指数及奇异性的双调和方程解的存在性

文中考虑了下面带奇异项的双调和方程 {?²u-μ/|x|^αμ =λg(x)μ+k(x)|μ|^q-2μ+|μ|^2*-2μ,x∈Ω, μ∈D₀^2,2(Ω), x∈∂Ω, 其中0∈Ω为R^N, N≥5中的有界区域, 0≤α, s < 4,2 < q < 2*(s) = 2(N-s)/N-4}, g(x), k(x) 为非负函数, 借助变分方法及嵌入映射D^2,2(R^N)→ L²*(R^N)的达到函数, 通过较精密的计算, 得到了上面方程解的存在性结果.