Mathematical Sciences: Analysis of Two-Scale Dilation Equations

数学科学：二尺度膨胀方程的分析

• 批准号: 9307601
• 负责人: Yang Wang
• 金额: $ 5.27万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-08-15 至 1997-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9307601&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Analysis; Two; Scale

9307601 Wang This research concerns two-scale dilation equations, which describe the two-scale behavior characteristic of scaling functions. These equations arise in a wide variety of applications, including wavelets, sub-division schemes for computer generation of curves and surfaces, and fractal geometry (iterated function systems - IFS). The principal investigator will develop techniques based on representations of the scaling function in terms of an infinite matrix product expansion, which can be used to establish an algorithm for constructing scaling functions, and to analyze their regularity, integrability, and dimensionality. Thus far, in order to ensure convergence of the infinite matrix product,a Contractivity Assumption has been imposed. This Assumption is stronger than necessary, and the PI is investigating weaker assumptions involving bounded semi-groups of matrices, and an average contractivity factor as determined by the Lyapunov exponent. This also leads to questions regarding the structure of bounded semi-groups of matrices. The PI has also developed techniques to study self-similar lattice tilings, which arises from multi-dimensional orthonormal wavelets. The PI intends to study the following: (1) surface generation: sliding window method; (2) existence of integrable solutions; (3) dilation equations with non-negative coefficients; (4) self-similar lattice tilings; (5) the Finiteness Conjecture of Daubechies and Lagarias. This activity had substantial potential for application in many areas, including compression algorithms. ***

小行星9307601 本文研究了双尺度伸缩方程，它描述了尺度函数的双尺度行为特征。 这些方程出现在各种各样的应用中，包括小波，用于计算机生成曲线和曲面的细分方案，以及分形几何（迭代函数系统- IFS）。主要研究者将开发基于无限矩阵乘积展开的尺度函数表示的技术，该技术可用于建立构建尺度函数的算法，并分析其规律性，可积性和维数。 到目前为止，为了确保无限矩阵乘积的收敛性，已经施加了收缩性假设。 该假设比必要的假设更强，PI正在研究涉及有界矩阵半群和由Lyapunov指数确定的平均收缩因子的较弱假设。 这也导致了关于矩阵的有界半群的结构的问题。 PI还开发了技术来研究自相似晶格平铺，这是从多维正交小波。 PI打算研究以下内容：（1）表面生成：滑动窗口方法;（2）可积解的存在性;（3）具有非负系数的膨胀方程;（4）自相似格平铺;（5）Daubechies和Lagarias的连续性猜想。 这一活动在许多领域，包括压缩算法，都有很大的应用潜力。 ***