关于L2(Rd)中仿射子空间小波标架的一个注记

研究了L2(Rd)的有限生成仿射子空间中小波标架的构造.证明了任意有限生成仿射子空间都容许一个具有有限多个生成元的Parseval小波标架,并且得到了仿射子空间是约化子空间的一个充分条件.对其傅里叶变换是一个特征函数的单个函数生成的仿射子空间,得到了与小波标架构造相关的投影算子在傅里叶域上的明确表达式,同时也给出了一些例子.     

伪仿射对偶框架

对于给定冗余的仿射框架X（Ф）={Ф1,…,ФL）,存在对偶Y（ψ）={ψ1,…,ψL）,这个对偶可能不是仿射框架,因为它不满足Bessel条件．这样的对偶我们称为伪仿射对偶,并给出伪仿射对偶框架的一种构造公式．     

类空和类时曲线的仿射性质研究

本文研究类空和类时曲线的中心仿射曲率,中心仿射挠率,曲线的曲率和挠率满足的关系以及两类曲线的正交标架和仿射标架之间的关系的问题.利用仿射空间和Minkowski空间中曲线的基本理论,讨论当类空和类时曲线的弧长与仿射弧长相同时,类空和类时曲线的仿射性质.根据得到的结论,通过变量代换讨论当类空和类时曲线的曲率κ(s)和挠率τ(s)满足τ(s)=aκ^λ(s)(a≠0,λ∈R)时,曲线的曲率所满足的特殊微分方程.     

平行三次形式的仿射超曲面

设Ａｎ 1是ｎ 1维仿射空间 ,Ｄ表示Ａｎ 1上的平坦联络 ,Ｍ是ｎ维光滑流形 ,ｘ:Ｍ→Ａｎ 1是一个非退化的仿射浸入 .对于Ｍ上的横截向量场ξ ,存在唯一的选择 ξ =1ｎ△ｘ(称为仿射法向量场 ) ,使得上述浸入是一个Ｂｌａｓｃｈｋｅ浸入 (见 [2 ]) .设 是此浸入由Ｄ在Ｍ上诱导的仿射联络 ,我们有 :ＤＸＹ= ＸＹ ｈ(Ｘ ,Ｙ) ξ　　　ＤＸξ=-ＳＸ这里Ｘ ,Ｙ ,Ｚ是Ｍ上的切向量场 ,ｈ是对称的双线性形式 ,由它可以定义Ｍ上的伪黎曼度量Ｇ(Ｇ =|Ｈ| - 1ｎ 2 ｈ ,Ｈ =ｄｅｔ(ｈｉｊ) ) ,称为Ｂｌａｓｃｈｋｅ度量 ,Ｓ称为Ｍ的形态算子 .若…     

乘积的平方性

本文证明了乘积 (f(x)=ax²+bx+c∈Z[x]是二次不可约多项式)在n充分大时不是平方数.       

仿射Weyl群上的Demazur乘积

Demazure乘积是定义在一般Coxeter群上的一类幺半群乘积.它自然地出现在李理论中的不同领域中.本文将研究仿射Weyl群上Demazur乘积.我们的主要结果是发现了它与有限Weyl群上的量子Bruhat图之间的一个紧密联系.作为应用,我们给出了仿射Weyl群最低双边胞腔元素之间Demazure乘积的显示表达式,并得到了最低双边胞腔元素的一般牛顿点以及Lusztig-Vogan映射的具体刻画.     

平方可积函数空间的仿射框架

推广了双正交小波的概念.引进了多尺度平移伪框架的概念.给出了它的塔式分解格式及其存在的条件.进而得到平方可积函数空间的函数仿射伪框架展式.     

仿射框架的新准则

仿射框架是小波理论中很基本的概念.熟知的关于仿射框架的Daubechies判别法引用了由绝对值| (ajω) (ajω lT)|算出的量来作判别.在本文建立的新判别法中引用了由(ajω) (ajω lT)的适当组合的代数和算出的量来作判别.当为偶函数时,新的判别法优于Daubechies判别法.     

仿射框架的新准则

仿射框架是小波理论中很基本的概念.熟知的关于仿射框架的Daubechies判别法引用了由绝对值|ψ(ajω)ψ(ajω+lT)|算出的量来作判别.在本文建立的新判别法中引用了由ψ(ajω)ψ(ajω+lT)的适当组合的代数和算出的量来作判别.当ψ为偶函数时,新的判别法优于Daubechies判别法.     

完备α度量的相对抛物型仿射球(英文)

摘要:设y:M→Rⁿ⁺¹是一个光滑连通流形到实仿射空间Rⁿ⁺¹的局部强凸浸入超曲面,而且是一个定义在区域Ω（?）Rⁿ上的严格凸函数x_n+1=f（x₁,x₂,…,x_n）的图.在α相对法化下,相对抛物型仿射球满足一个四阶非线性偏微分方程组.本文证明了这类抛物型仿射球的一个新的Bernstein性质.     

Affine frames, quasi-affine frames, and their duals

The notion of quasi-affine frame was recently introduced by Ron and Shen [9] in order to achieve shift-invariance of the discrete wavelet transform. In this paper, we establish a duality-preservation theorem for quasi-affine frames. Furthermore, the preservation of frame bounds when changing an affine frame to a quasi-affine frame is shown to hold without the decay assumptions in [9]. Our consideration leads naturally to the study of certain sesquilinear operators which are defined by two affine systems. The translation-invariance of such operators is characterized in terms of certain intrinsic properties of the two affine systems. This revised version was published online in August 2006 with corrections to the Cover Date.     

Partial affine system–based frames and dual frames

In this paper, we introduce the notion of partial affine system that is a subset of an affine system. It has potential applications in signal analysis. A general affine system has been extensively studied; however, the partial one has not. The main focus of this paper is on partial affine system–based frames and dual frames. We obtain a necessary condition and a sufficient condition for a partial affine system to be a frame and present a characterization of partial affine system–based dual frames. Some examples are also provided.     

Affine dual frames and Extension Principles

In this work we provide three new characterizations of affine dual frames constructed from refinable functions. The first one is similar to Daubechies et al. (2003) [10, Proposition 5.2] but without any decay assumptions on the generators of a pair of affine systems. The second one reveals the geometric significance of the Mixed Fundamental function and the third one shows that the Mixed Oblique Extension Principle actually characterizes dual framelets. We also extend recent results on the characterization of affine Parseval frames obtained in Stavropoulos (2012) [27, Theorem 2.3].     

Subspace dual super wavelet and Gabor frames

Due to its potential applications in multiplexing techniques, the study of superframes has interested some researchers. This paper addresses dual super wavelet and Gabor frames in the subspace setting. We obtain a basic-equation characterization for subspace dual super wavelet and Gabor frames. In addition, applying this characterization, we derive a procedure that allows for constructing subspace dual super wavelet frames based on certain subspace dual super Gabor frames, and vice versa. Our results are new even in L²(R;C ^L ) setting.     

Parameterizations of Masks for Tight Affine Frames with Two Symmetric/Antisymmetric Generators

Parameterizations of FIR orthogonal systems are of fundamental importance to the design of filters with desired properties. By constructing paraunitary matrices, one can construct tight affine frames. In this paper we discuss parameterizations of paraunitary matrices which generate tight affine frames with two symmetric/antisymmetric generators (framelets). Based on the parameterizations, several symmetric/antisymmetric framelets are constructed.     

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.     

The Matrix-Valued Riesz Lemma and Local Orthonormal Bases in Shift-Invariant Spaces

We use the matrix-valued Fejér–Riesz lemma for Laurent polynomials to characterize when a univariate shift-invariant space has a local orthonormal shift-invariant basis, and we apply the above characterization to study local dual frame generators, local orthonormal bases of wavelet spaces, and MRA-based affine frames. Also we provide a proof of the matrix-valued Fejér–Riesz lemma for Laurent polynomials.     

Affine systems inL 2 (ℝ d ) II: Dual systems

The fiberization of affine systems via dual Gramian techniques, which was developed in previous papers of the authors, is applied here for the study of affine frames that have an affine dual system. Gramian techniques are also used to verify whether a dual pair of affine frames is also a pair of bi-orthogonal Riesz bases. A general method for a painless derivation of a dual pair of affine frames from an arbitrary MRA is obtained via the mixed extension principle. This work was partially sponsored by the National Science Foundation under Grants DMS-9102857, DMS-9224748, and DMS-9626319, by the United States Army Research Office under Contracts DAAL03-G-90-0090, DAAH04-95-1-0089, and by the Strategic Wavelet Program Grant from the National University of Singapore.     

Affine Density,Frame Bounds,and the Admissibility Condition for Wavelet Frames

For a large class of irregular wavelet frames we derive a fundamental relationship between the affine density of the set of indices, the frame bounds, and the admissibility constant of the wavelet. Several implications of this theorem are studied. For instance, this result reveals one reason why wavelet systems do not display a Nyquist phenomenon analogous to Gabor systems, a question asked in Daubechies' Ten Lectures book. It also implies that the affine density of the set of indices associated with a tight wavelet frame has to be uniform. Finally, we show that affine density conditions can even be used to characterize the existence of wavelet frames, thus serving, in particular, as sufficient conditions.