On the computation of scaling coefficients of Daubechies' wavelets

In the present paper, Daubechies' wavelets and the computation of their scaling coefficients are briefly reviewed. Then a new method of computation is proposed. This method is based on the work [7] concerning a new orthonormality condition and relations among scaling moments, respectively. For filter lengths up to 16, the arising system can be explicitly solved with algebraic methods like Gröbner bases. Its simple structure allows one to find quickly all possible solutions.     

Symbolic Computation for Moments and Filter Coefficients of Scaling Functions

Algebraic relations between discrete and continuous moments of scaling functions are investigated based on the construction of Bell polynomials. We introduce families of scaling functions which are parametrized by moments. Filter coefficients of scaling functions and wavelets are computed with computer algebra methods (in particular Gröbner bases) using relations between moments. Moreover, we propose a novel concept for data compression based on parametrized wavelets.Received December 15, 2003     

Compactly Supported Wavelets Through the Classical Umbral Calculus

We explore compactly supported scaling functions of wavelet theory by means of classical umbral calculus as reformulated by Rota and Taylor. We set a theory of orthonormal scaling umbra which leads to a very simple and elementary proof of Lawton's theorem for umbrae. When umbrae come from a wavelet setting, we recover the usual Lawton condition for the orthonormality of the integer translates of a scaling function.     

小波分析中一个逆问题

本文证明:如果来自多尺度分析(伸缩因子为矩阵)的小波是标准正交的,那么相对应的尺度函数也是标准正交的,其中函数f_s(x)∈L~2(R~n)(s=1,2,…,r,r是正整数)的标准正交性是指f_s(x)的整平移所构成的函数族为L~2(R~n)的标准正交系。结果表明,如果我们想从多尺度分析出发构造正交小波,那么该多尺度分析必须有正交尺度函数。     

Propagation of round-off error in a model of quadratic rational rotations

We investigate the propagation of round-off error for a discrete map modeling a one-dimensional linear oscillator viewed stroboscopically in phase space, with uniform, non-dissipative round-off. The probability P(r,t) of a net displacement r during t time steps can be reduced, essentially, to a weighted sum over contributions from a small number of infinite scaling sequences of periodic orbits. We show that the successive members of each scaling sequence can be built up by application of a set of substitution rules. This implies recursion relations, not only for the geometry of the orbits, but also for P(r,t) and its moments, allowing these quantities to be calculated exactly as algebraic numbers. For asymptotically large t, the moments have power-law increase, modulated by log-periodic or (in one particularly interesting case) log-quasi-periodic oscillations.     

The factorization of generalized Nevanlinna functions and the invariant subspace property

The well-known invariant subspace property of selfadjoint relations (multi-valued operators) in Pontryagin spaces is shown to be equivalent to the factorization property of (scalar) generalized Nevanlinna functions. This connection is established by a new realization for generalized Nevanlinna functions explicitly reflecting this connection. Combining this result with the new function-theoretic proof for the factorization property of generalized Nevanlinna functions contained in Wietsma (2018) immediately yields a new proof for the invariant subspace property of selfadjoint relations in Pontryagin spaces.     

Quadrature Formulas Based on the Scaling Function

The scaling function corresponding to the Daubechies wavelet with two vanishing moments is used to derive new quadrature formulas. This scaling function has the smallest support among all orthonormal scaling functions with the properties M ₂ = M ₁ ² and M ₀ = 1. So, in this sense, its choice is optimal. Numerical examples are given.This work was partially supported by DFG grant GR 1777/2, by the Grant No 201/01/1200 of the CSF, by the grant MSMT 113200007 and by the grant IGS 116/5130/1 of FP TUL.     

On the fast approximation of some nonlinear operators in nonregular wavelet spaces

We focus our attention on the approximation of some nonlinear operators in adapted wavelet spaces. We show the interest of the construction of scaling functions with a large number of zero moments. We present the convergence estimate of an algorithm based on paraproducts for the approximation of nonlinear operators using wavelets connected to scaling functions with zero moments. Numerical tests are performed on univariate examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

具有一定消失矩的优化线性相位滤波器

利用消失矩特性和编码误差最小,给出了一类有限长线性相位的双正交小波滤波器组BNVF的构造方法,BNVF的综合低通和分解高通的系数为二进分数,将其用于图像的分解与重构时,可避免一些乘法运算．三组BNVF用于图像变换编码时,其压缩性能不亚于Cohen,Daubechies等构造的9／7步滤波器组CDF-9／7,且计算复杂度更低．     

混合体积、对数凹序列和B样条函数

本文主要通过样条函数方法研究与之相关的离散几何学和组合学问题.在离散几何学方面主要考虑超立方体切面(cube slicing)体积和混合体(mixed volume)的样条表示,利用B样条函数的几何解释,将超立方体切面问题转化为与之等价的样条函数问题,分别给出Laplace和P′olya关于超立方体切面定理的样条证明,将样条函数与混合体积联系起来,给出一类混合体积的样条解释.利用这种解释可以得到一类具有对数凹性质的组合序列,从而部分地回答了Schmidt和Simion所提出的关于混合体积的公开问题.在组合数学方面主要考虑多种组合多项式与样条函数的关联以及组合序列对数凹性质的样条方法研究.本文借助丰富的样条函数理论,不但验证了离散几何学和组合数学中很多现有的结果,而且得到了一系列离散数学对象的新性质,建立了离散数学问题与具有连续性特质的样条函数之间的内在联系.     

The Asymptotic Behavior of Certain Birth Processes

We describe a connection between discrete birth process and a certain family of multivariate interpolation polynomials. This enables us to compute all asymptotic moments of the birth process, generalizing previously known results for the mean and variance. Received July 15, 2004     

Discrete-Euclidean operations

In this paper, we study the relationship between Euclidean and discrete space. We study discrete operations based on Euclidean functions: discrete smooth scaling and discrete-continuous rotation. Conversely, we study Euclidean operations based on discrete functions: the discrete based simplification, the Euclidean-discrete union and the Euclidean-discrete co-refinement. These operations operate partly in discrete, and partly in continuous space. Especially for the discrete smooth scaling operation, we provide error bounds when such different operations are chained.     

Scaling transformation and probability distributions for financial time series

The price of financial assets are, since [Bachelier L. Annales de l'Ecole Normale Supérieure 1900;3:XVII:21–86], considered to be described by a (discrete or continuous) time sequence of random variables, i.e., a stochastic process. Sharp scaling exponents or unifractal behavior of such processes has been reported in several works [Mandelbrot BB. J Business 1963;36:394–419; Peters EE. Chaos and order in the capital markets. New York: Wiley, 1991; Mantegna RN, Stanley HE. Nature 1995;376:46–49; Evertsz CJG. Fractals. 1995;3:609–616; Bouchaud JP, Potters M. Théorie des risques financiers. Aléa Saclay, 1997]. In this paper we investigate the question of scaling transformation of price processes by establishing a new connection between non-linear group theoretical methods and multifractal methods developed in mathematical physics. Using two sets of financial chronological time series, we show that the scaling transformation is a non-linear group action on the moments of the price increments. Its linear part has a spectral decomposition that puts in evidence a multifractal behavior of the price increments.     

Fast scaling algorithms for M-convex function minimization with application to the resource allocation problem

M-convex functions, introduced by Murota (Adv. Math. 124 (1996) 272; Math. Prog. 83 (1998) 313), enjoy various desirable properties as “discrete convex functions.” In this paper, we propose two new polynomial-time scaling algorithms for the minimization of an M-convex function. Both algorithms apply a scaling technique to a greedy algorithm for M-convex function minimization, and run as fast as the previous minimization algorithms. We also specialize our scaling algorithms for the resource allocation problem which is a special case of M-convex function minimization.     

Hausdorff's moment problem and expansions in Legendre polynomials

A new proof is given for Hausdorff's condition on a set of moments which determines when the function generating these moments is in L². The proof uses Legendre polynomials and their discrete extensions found by Tchebychef. Then an extension is given to a weighted L² space using Jacobi polynomials and their discrete extensions.     

Truncated random variables with application to random surfaces

There has been considerable interest in obtaining discrete results for random surfaces. Standard results have been published in journals of physics or engineering which have emphasised the applications. This paper gives a detailed background of the mathematical methods needed so that the central connection, namely truncated random variables, between these standard results can be understood. Distributions of discrete peak measures are obtained from the distributions of discrete profile measures of a random Gaussian surface by applying results for the distributions of truncated random variables. This enable the moments to be obtained from known results for the truncated distributions.     

Entropic Moments and Domains of Attraction on Countable Alphabets

Modern information theory is largely developed in connection with random elements residing in large, complex, and discrete data spaces, or alphabets. Lacking natural metrization and hence moments, the associated probability and statistics theory must rely on information measures in the form of various entropies, for example, Shannon’s entropy, mutual information and Kullback–Leibler divergence, which are functions of an entropic basis in the form of a sequence of entropic moments of varying order. The entropicmoments collectively characterize the underlying probability distribution on the alphabet, and hence provide an opportunity to develop statistical procedures for their estimation. As such statistical development becomes an increasingly important line of research in modern data science, the relationship between the underlying distribution and the asymptotic behavior of the entropic moments, as the order increases, becomes a technical issue of fundamental importance. This paper offers a general methodology to capture the relationship between the rates of divergence of the entropic moments and the types of underlying distributions, for a special class of distributions. As an application of the established results, it is demonstrated that the asymptotic normality of the remarkable Turing’s formula for missing probabilities holds under distributions with much thinner tails than those previously known.     

Global attractivity in delayed Cohen–Grossberg neural network models

In this paper, we investigate the global attractivity of Cohen–Grossberg neural network models with connection time delays for both discrete and distributed cases via the Lyapunov functional method. Without assuming the monotonicity and differentiability of activation functions and the symmetry of connection matrix, we establish three new sufficient conditions for the global exponential stability of a unique equilibrium for the delayed Cohen–Grossberg neural network no matter whether the connection time delay is of discrete type or distributed type. In particular, all the three new criteria are independent of time delays and do not include one another. To demonstrate the differences and features of the new stability criteria, several examples are discussed to compare the present results with the existing ones.     

Moment solution to an urn model

A subset of Bernard's RD-model (replenishment-depletion) is considered from the viewpoint of the calculus of finite differences. The most general case is considered and includes an urn with balls of many colors, each color being replenished either deterministically or stochastically. Factorial moment generating functions (fmgfs) are employed to define probability generating functions. A new result is given for the two color case defining the fmgf and probability generating function (with probabilities) when the replenishments are positive valued random variables with given factorial moments. This result involves beta integral transforms defining a manifold of discrete distributions. Particular cases relate to hypergeometric discrete distributions.This research was partly supported by Martin Marietta Energy Systems, Inc., under contract DE-AC05-84OR21400 with the U.S. Department of Energy.     

带小波函数的Cauchy主值积分的数值计算

1 引言 众所周知,小波方法在信号处理和图像处理方面发挥了举世瞩目的成就。近年来人们研究小波方法在数值分析方面的应用。期望在数值求解微分方程和积分方程方面发挥良好的作用。本文研究带有小波函数的Cauchy主值积分 的数值计算方法,其中Φ(x)是紧支撑的尺度函数。这是数值求解积分方程的核心问题之一。 1.l 多分辩分析 空间L~2(R)中的一个多分辩分析是这样的闭子空间列{V_j},它满足下列条件 1) 2) 3) 4)存在尺度函数,使构成V_o的Riesz基,从而也存在序列使满足双尺度方程