多元正交小波基

本文构造了一组Haar型的多元正交小波基,它们不能用张量积的方法产生.     

High-Order Orthonormal Scaling Functions and Wavelets Give Poor Time-Frequency Localization

For a fairly general class of orthonormal scaling functions and wavelets with regularity exponents n, we prove that the areas of the time-frequency windows tend to infinity as n → ∞. This class includes those of Battle-Lemarie and Daubechies. In addition, if the scaling functions have at least asymptotic linear phase, then we prove that they converge to the "sinc" function and their corresponding orthonormal wavelets converge to the "difference" of two sinc functions.     

Non-separable Lattices,Gabor Orthonormal Bases and Tilings

Let \(K\subset {\mathbb {R}}^d\) be a bounded set with positive Lebesgue measure. Let \(\Lambda =M({\mathbb {Z}}^{2d})\) be a lattice in \({\mathbb {R}}^{2d}\) with density dens\((\Lambda )=1\). It is well-known that if M is a diagonal block matrix with diagonal matrices A and B, then \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis for \(L^2({\mathbb {R}}^d)\) if and only if K tiles both by \(A({\mathbb {Z}}^d)\) and \(B^{-t}({\mathbb {Z}}^d)\). However, there has not been any intensive study when M is not a diagonal matrix. We investigate this problem for a large class of important cases of M. In particular, if M is any lower block triangular matrix with diagonal matrices A and B, we prove that if \({\mathcal {G}}(|K|^{-1/2}\chi _K, \Lambda )\) is an orthonormal basis, then K can be written as a finite union of fundamental domains of \(A({{\mathbb {Z}}}^d)\) and at the same time, as a finite union of fundamental domains of \(B^{-t}({{\mathbb {Z}}}^d)\). If \(A^tB\) is an integer matrix, then there is only one common fundamental domain, which means K tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domain is also possible. We also provide a constructive way for forming a Gabor window function for a given upper triangular lattice. Our study is related to a Fuglede’s type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.

     

Adapted Bases of Time-Frequency Local Cosines

We develop and analyze a best basis algorithm for orthonormal bases of local cosines which satisfy a uniform bound on their time-frequency concentration. All waveforms are obtained from three elementary window functions by shifts, rescaling, and modulation. For a discrete signal of length N, the complexity is of order N(log N)².     

Korobov Spaces with Respect to Digital Orthonormal Bases

 For an orthonormal basis (ONB) of we define classes of functions according to the order of decay of the Fourier coefficients with respect to the considered ONB . The rate is expressed in the real parameter α. We investigate the following problem: What is the order of decay, if any, when we consider with respect to another ONB ? If the function is expressable as an absolutely convergent Fourier series with respect to , we give bounds for the new order of decay, which we call . Special attention is given to digital orthonormal bases (dONBs) of which the Walsh and Haar systems are examples treated in the present paper. Bounding intervals and in several cases explicit values for are given for the case of dONBs. An application to quasi-Monte Carlo numerical integration is mentioned.     

Haar Type Orthonormal Wavelet Bases in R2

K.-H. Grochenig and A. Haas asked whether for every expanding integer matrix A ∈ M_n(ℤ) there is a Haar type orthonormal wavelet basis having dilation factor A and translation lattice ℤⁿ. They proved that this is the case when the dimension n = 1. This article shows that this is also the case when the dimension n = 2.     

Korobov Spaces with Respect to Digital Orthonormal Bases

 For an orthonormal basis (ONB) of we define classes of functions according to the order of decay of the Fourier coefficients with respect to the considered ONB . The rate is expressed in the real parameter α. We investigate the following problem: What is the order of decay, if any, when we consider with respect to another ONB ? If the function is expressable as an absolutely convergent Fourier series with respect to , we give bounds for the new order of decay, which we call . Special attention is given to digital orthonormal bases (dONBs) of which the Walsh and Haar systems are examples treated in the present paper. Bounding intervals and in several cases explicit values for are given for the case of dONBs. An application to quasi-Monte Carlo numerical integration is mentioned. (Received 21 February 2000; in revised form 19 October 2000)     

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.     

Control Variables with Known Mean and Variance

In computer simulation experiments, when applying the method of control variates, it is useful to know the covariance matrix as well as the means of the control variates. Most control variates suggested in the literature are sample means whose true mean is known but whose covariance matrix is not. Such variates are shown not to be the best, and that by the use of standardised sums rather than sample means, control variates are obtained with known covariances as well as means. Linear combinations of these variates are shown to be best in that more complicated functions of these variates do not achieve additional increase in estimate accuracy. Numerical illustrations are given.     

Random Probability Measures with Given Mean and Variance

This article describes several natural methods of constructing random probability measures with prescribed mean and variance, and focuses mainly on a technique which constructs a sequence of simple (purely discrete, finite number of atoms) distributions with the prescribed mean and with variances which increase to the desired variance. Basic properties of the construction are established, including conditions guaranteeing full support of the generated measures, and conditions guaranteeing that the final measure is discrete. Finally, applications of the construction method to optimization problems such as Plackett's Problem are mentioned, and to experimental determination of average-optimal solutions of certain control problems.     

Some Methods for Constructing Nonseparable,Orthonormal, Compactly Supported Wavelet Bases

We first show that by combining monodimensional filter banks one can obtain nonseparable filter banks. We then give necessary conditions for these filter banks to generate orthonormal and regular wavelets. Finally, we establish that some of these filter banks lead to arbitrarily smooth, nonseparable, orthonormal, compactly supported wavelet bases.     

Orthonormal Sequences in L 2(R d ) and Time Frequency Localization

We prove that there does not exist an orthonormal basis {b _n } for L ²(R) such that the sequences {μ(b _n )}, {m([^(b_n)])}\{\mu(\widehat{b_{n}})\} , and {D(b_n)D([^(b_n)])}\{\Delta(b_{n})\Delta(\widehat{b_{n}})\} are bounded. A higher dimensional version of this result that involves generalized dispersions is also obtained. The main tool is a time-frequency localization inequality for orthonormal sequences in L ²(R ^d ). On the other hand, for d>1 we construct a basis {b _n } for L ²(R ^d ) such that the sequences {μ(b _n )}, {m([^(b_n)])}\{\mu(\widehat{b_{n}})\} , and {D(b_n)D([^(b_n)])}\{\Delta(b_{n})\Delta(\widehat{b_{n}})\} are bounded.     

Alternative Local Discriminant Bases Using Empirical Expectation and Variance Estimation

This article proposes alternative discriminant measures for selecting the best basis for classification purposes among a large collection of orthonormal bases organized in a binary tree structure. A generalization of the local discriminant basis algorithm of Saito and Coifman is constructed. The success of these new methods is evaluated and compared to earlier methods in experiments.     

几个由积分形成的数列的极限

围绕着若干个由积分定义的数列展开讨论,求出了这些数列的极限,并指出它们是一些已知的数列极限的推广。     

双变量参数联合函数的条件均值和条件方差(英文)

联合函数是指连接单变量边际分布的多变量函数.联合函数由Sklar(1959)在概率测度空间的内容时引入的.本文主要对边际分布是标准正态分布函数U(0,1)的Farlie-Gum-bel-Morgenstern和Gumbel-Hougaard这两个双变量参数联合函数进行研究,我们得到了他们密度函数的基本性质并导出了他们的条件均值和条件方差.另外,本文还给出了不同参数的条件均值和条件方差的相应图示,并进行了对比和解释.     

时间序列期望和方差函数的半参数估计方法

以二阶差分序列为基础,针对相关系数平稳时间序列,给出了其期望函数与方差函数的半参数估计形式.核方法的运用使得无需假定非周期项的具体形式就能给出相应估计结果,避免了可能出现的假定错误.进一步,通过选取不同的判定函数,得到相应方差函数的估计结果,并对此类方法进行了理论上的总结.最后通过仿真实验验证了所提出方法的有效性.     

指令性抽样下总体均值和方差的估计及其应用

指令性抽样下的样本往往不具有代表性,因此仅用它们来推断总体将是不适合的,这篇文章基于观察信息,利用概率统计方法将没有发出调查指令的样本信息补充出来,然后利用观察信息和补充信息一起来对总体进行推断.具体我们给出了总体均值和方差参数估计的迭代公式,并给出它们在经济犯罪调查和流行病调查中的应用.     

非线性均值方差模型的贝叶斯数据删除统计诊断

文章在非线性均值方差模型框架下基于K-L距离研究贝叶斯数据删除影响的统计诊断问题,通过应用Gibbs抽样和MH算法估计贝叶斯数据删除影响诊断统计量.随机模拟研究和红鳟鲑鱼数据的数值例子说明该诊断方法的可行性.