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.

该项目涉及多维小波展开数学理论的谐波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良好时频局部化等特性的小波。尽管这一领域取得了重大进展，但大多数研究都集中在各向同性理论上，非各向同性小波理论远远落后。本项目的主要研究方向之一是开发用于大类别非各向同性膨胀膨胀的良好定域正交小波的构造技术。一个密切相关的补充主题是识别非各向同性膨胀，其中良好定域小波的构造是不可能的。该项目的另一个方向是研究与膨胀膨胀相关的经典函数空间的非各向同性类似物。更一般地说，这个项目代表了小波分析的工作，小波分析是谐波分析中的一种强大的技术。与已有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都具有广泛的应用潜力。该项目的主要目标是整合多维小波理论的研究和教育活动，使非各向同性小波的进一步应用成为可能。