Mathematical Sciences: Adaptive Frame Decompositions

数学科学：自适应框架分解

• 批准号: 9307655
• 负责人: John Gilbert
• 金额: $ 10.35万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-15 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9307655&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Adaptive; Frame; Decompositions

9307655 Gilbert This project will develop adaptive tools for decomposing function spaces that extend methods from Fourier analysis and wavelet theory. Specific functions are to be analyzed through the selection of a particular wavelet well-adapted to the function. Criteria for choosing the matched wavelet come from the specific application which creates the function. Families of analyzing wavelets will be designed having specific smoothness and decay properties needed for establishing such optimization criteria. Frame decompositions will also be studied in vector-valued settings in relation to extensions of the affine group and corresponding representation theory. This work will draw on a theory, developed recently which relates the generalized Cauchy- Riemann systems to unitarizable representations of groups containing the affine group. A particularly relevant example, previously studied by Torresani, is the semidirect product of the affine group with the Heisenberg group. Corresponding frame decompositions will incorporate aspects of both the short-time Fourier transform and wavelet expansions. Mathematical research focusing on wavelets and frame decompositions has its roots in classical harmonic analysis but derives its recent impetus from studies in signal processing. Here, the wavelet concept has proved remarkably effective in analyzing signals composed of both stationary and transient features. Traditional Fourier methods have not been effective tools in these studies. Applications to research on acoustics, data compression and partial differential equations has already proven the worth of the newly developing ideas. ***

小吉尔伯特9307655 这个项目将开发用于分解函数空间的自适应工具，这些工具扩展了傅立叶分析和小波理论的方法。 具体的功能是通过选择一个特定的小波很好地适应功能进行分析。 选择匹配小波的标准来自创建函数的特定应用。 分析小波的家庭将被设计为具有特定的光滑性和衰减特性，需要建立这样的优化标准。 框架分解也将研究在向量值设置有关的仿射群和相应的表示理论的扩展。 这项工作将借鉴一个理论，最近开发的广义柯西-黎曼系统的unitarizable表示的群体包含仿射群。 一个特别相关的例子，以前研究的Torresani，是半直积的仿射群与海森堡群。 相应的帧分解将包含短时傅立叶变换和小波展开的方面。 专注于小波和框架分解的数学研究起源于经典的谐波分析，但最近的动力来自信号处理的研究。在这里，小波的概念已被证明是非常有效的分析信号组成的平稳和瞬态特征。 传统的傅立叶方法在这些研究中已经不是有效的工具。 在声学、数据压缩和偏微分方程研究中的应用已经证明了新发展思想的价值。 ***