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: 0653881
• 负责人: Marcin Bownik
• 金额: $ 11.96万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0653881&HistoricalAwards=false
• 关键词: Multidimensional; wavelets; non; isotropic; function

The project involves education and research activities in harmonic analysis concerning the mathematical theory of multi-dimensional wavelet expansions. One of the fundamental problems of the subject is how to construct wavelets with desired properties such as good time-frequency localization. Despite significant progress in this area, the majority of research has been concentrated on isotropic theory andnon-isotropic wavelet theory lags far behind. One of the main research directions of the project is the development of techniques for construction of well-localized orthogonal wavelets for large classesof non-isotropic expansive dilations. A closely related complementary topic is the identification of non-isotropic dilations for which the construction of well-localized wavelets is not possible.Another direction of the project is to study non-isotropic analogues of classical function spaces associated to expansive dilations. More generally, the project represents work on wavelet analysis which is a powerful technique in harmonic analysis. Non-isotropic wavelet theory in higher dimensions has a potential for wide applications both in pure analysis and more applied areas of signal and image processing comparable to that of already well established isotropic wavelet theory. The main goal of the project is to integrate research and education activities in the multidimensional wavelet theory which could make further applications of non-isotropic wavelets possible.

该项目涉及有关多维小波展开数学理论的调和分析的教育和研究活动。该学科的基本问题之一是如何构造具有所需特性（例如良好的时频局部化）的小波。尽管该领域取得了重大进展，但大多数研究集中在各向同性理论上，非各向同性小波理论远远落后。该项目的主要研究方向之一是开发大类非各向同性膨胀扩张的定域正交小波构造技术。一个密切相关的补充主题是识别非各向同性扩张，对于非各向同性扩张，构造良好局域化的小波是不可能的。该项目的另一个方向是研究与扩张扩张相关的经典函数空间的非各向同性类似物。更一般地说，该项目代表了小波分析的工作，小波分析是调和分析中的一种强大技术。与已经完善的各向同性小波理论相比，高维非各向同性小波理论在纯分析以及信号和图像处理的更多应用领域具有广泛的应用潜力。该项目的主要目标是将多维小波理论的研究和教育活动结合起来，使非各向同性小波的进一步应用成为可能。