Wavelet para-bases and sampling numbers in function spaces on domains

This paper deals with wavelet frames (para-bases), local polynomial reproducing formulas, and sampling numbers in function spaces on arbitrary and on E-thick domains in Euclidean n-space. In an Appendix we collect some recent instruments for corresponding function spaces on Euclidean n-space.     

Balanced biorthogonal scaling vectors using fractal function macroelements on

Geronimo, Hardin et al. have previously constructed orthogonal and biorthogonal scaling vectors by extending a spline scaling vector with functions supported on [0,1]. Many of these constructions occurred before the concept of balanced scaling vectors was introduced. This paper will show that adding functions on [0,1] is insufficient for extending spline scaling vectors to scaling vectors that are both orthogonal and balanced. We are able, however, to use this technique to extend spline scaling vectors to balanced, biorthogonal scaling vectors, and we provide two large classes of this type of scaling vector, with approximation order two and three, respectively, with two specific constructions with desirable properties in each case. The constructions will use macroelements supported on [0,1], some of which will be fractal functions.     

Locally supported rational spline wavelets on a sphere

In this paper we construct certain continuous piecewise rational wavelets on arbitrary spherical triangulations, giving explicit expressions of these wavelets. Our wavelets have small support, a fact which is very important in working with large amounts of data, since the algorithms for decomposition, compression and reconstruction deal with sparse matrices. We also give a quasi-interpolant associated to a given triangulation and study the approximation error. Some numerical examples are given to illustrate the efficiency of our wavelets.

     

Wavelet-based limiting performance analysis of mechanical systems subject to transient disturbances

Wavelets are used as the basis functions for control forces in the limiting performance analysis of a mechanical system subject to transient loading. The limiting performance problem, an open-loop control problem, can then be treated as a mathematical programming problem. A numerical example is presented to illustrate the effectiveness of using wavelets. Conventional approximate representations of the control function are interpreted in terms of the scaling function and multiresolution.     

SPECTRAL APPROXIMATION ORDERS OF MULTIDIMENSIONAL NONSTATIONARY BIORTHOGONAL SEMI-MULTIRESOLUTION ANALYSIS IN SOBOLEV SPACE

Subdivision algorithm （Stationary or Non-stationary） is one of the most active and exciting research topics in wavelet analysis and applied mathematical theory. In multidimensional non-stationary situation, its limit functions are both compactly supported and infinitely differentiable. Also, these limit functions can serve as the scaling functions to generate the multidimensional non-stationary orthogonal or biorthogonal semi-multiresolution analysis （Semi-MRAs）. The spectral approximation property of multidimensional non-stationary biorthogonal Semi-MRAs is considered in this paper. Based on nonstationary subdivision scheme and its limit scaling functions, it is shown that the multidimensional nonstationary biorthogonal Semi-MRAs have spectral approximation order r in Sobolev space H^s（R^d）, for all r ≥ s ≥ 0.     

Multifractal Formalism for Self-Similar Functions Under the Action of Nonlinear Dynamical Systems

We study functions which are self-similar under the action of some nonlinear dynamical systems. We compute the exact pointwise H{?}lder regularity, then we determine the spectrum of singularities and the Besov ``smoothness' index, and finally we prove the multifractal formalism. The main tool in our computation is the wavelet analysis. October 1, 1996. Date revised: May 13, 1997. Date re-revised: January 10, 1998. Date accepted: February 27, 1998.     

Shadow block iteration for solving linear systems obtained from wavelet transforms

Matrices resulting from wavelet transforms have a special “shadow” block structure, that is, their small upper left blocks contain their lower frequency information. Numerical solutions of linear systems with such matrices require special care. We propose shadow block iterative methods for solving linear systems of this type. Convergence analysis for these algorithms are presented. We apply the algorithms to three applications: linear systems arising in the classical regularization with a single parameter for the signal de-blurring problem, multilevel regularization with multiple parameters for the same problem and the Galerkin method of solving differential equations. We also demonstrate the efficiency of these algorithms by numerical examples in these applications.     

Circulant Wavelet Preconditioners for Solving Elliptic Differential Equations and Boundary Integral Equations

In this paper is discussed solving an elliptic equation and a boundary integral equation of the second kind by representation of compactly supported wavelets. By using wavelet bases and the Galerkin method for these equations, we obtain a stiff sparse matrix that can be ill-conditioned. Therefore, we have to introduce an operator which maps every sparse matrix to a circulant sparse matrix. This class of circulant matrices is a class of preconditioners in a Banach space. Based on having some properties in the spectral theory for this class of matrices, we conclude that the circulant matrices are a good class of preconditioners for solving these equations. We called them circulant wavelet preconditioners (CWP). Therefore, a class of algorithms is introduced for rapid numerical application.     

多层次统计-动力建模及其应用

该文将子波变换技术和平衡态动力理论结合起来，提出了以气候突变点数为核心的代层次气候建模技术。该技术不仅具有纯粹的动力学意义，而且模式的物理意义十分清楚 ，计算简单。     

多频率多函数小波

本文把通过方向多分辨分析构造的由一个函数生成的多频率小波推广到由有限个函数生成的多频率小波，给出由n2j1 j2个函数φ1,…φn,ψ1,…ψn(2j1 j2-1)的平移生成Vj(l)空间的Riesz基的充分必要条件．