Multidimensional Wavelets in Non-Isotropic Function Spaces

非各向同性函数空间中的多维小波

• 批准号: 0200080
• 负责人: Marcin Bownik
• 金额: $ 8.02万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-15 至 2004-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200080&HistoricalAwards=false
• 关键词: Multidimensional; Wavelets; Non; Isotropic; Function

This project investigates the areas of harmonic analysis andwavelets concerning the mathematical theory of multi-dimensional wavelet expansions. One of the fundamentalproblems of the subject is how to construct higher dimensionalwavelet bases with desired characteristics, e.g., wavelets withgood time-frequency properties. These areas of research haveseen significant progress due to the contributions of I.Daubechies, R. Coifman, and Y. Meyer, along with many others.However, the majority of research has been concentrated onisotropic theory, leaving many questions involving non-isotropic wavelet theory unanswered. The proposer willinvestigate this theory of non-isotropic wavelets from threedirections. The first is to study non-isotropic analogues ofthe standard function spaces associated with expansive dilations.In particular, to examine characterization by wavelet expansionsof Calderon-Zygmund operators associated with non-isotropicdilations. The second is to construct orthogonal wavelets withgood time-frequency localization for large classes of non-isotropic expansive dilations. The third is to identify non-isotropic expansive dilations for which the construction ofwell-localized wavelets is impossible.More generally, this proposal represents work on waveletanalysis which is a powerful technique in harmonic analysis.This technique has produced wide-ranging applications to signaland image processing, such as the JPEG 2000 image compressionsystem. It is expected that further research on multi-dimensional non-isotropic wavelets will continue to produce manyother contributions in pure and applied mathematics.

该项目研究调和分析和小波领域有关多维小波展开的数学理论。本课题的基本问题之一是如何构造具有所需特性的高维小波基，例如具有良好时频特性的小波。由于 I.Daubechies、R. Coifman 和 Y. Meyer 等人的贡献，这些研究领域取得了重大进展。然而，大多数研究都集中在各向同性理论上，留下了许多涉及非各向同性小波的问题理论没有答案。提出者将从三个方向研究非各向同性小波理论。第一个是研究与膨胀膨胀相关的标准函数空间的非各向同性类似物。特别是，通过与非各向同性膨胀相关的 Calderon-Zygmund 算子的小波展开来检查表征。第二个是针对大类非各向同性膨胀膨胀构造具有良好时频局部化的正交小波。第三个是识别非各向同性扩张扩张，对于这种扩张，构造良好局域化的小波是不可能的。更一般地说，这个建议代表了小波分析的工作，这是调和分析中的一种强大技术。该技术在信号和图像领域产生了广泛的应用处理，例如 JPEG 2000 图像压缩系统。预计对多维非各向同性小波的进一步研究将继续在纯数学和应用数学中产生许多其他贡献。