Mathematical Sciences: Adaptive Frame Decompositions

这个项目将开发自适应工具来分解函数空间，扩展傅里叶分析和小波理论的方法。具体的函数是通过选择与函数相适应的特定小波来分析的。选择匹配小波的标准来自于创建函数的具体应用。分析小波的族将被设计成具有建立这种优化准则所需的特定平滑和衰减特性。框架分解也将在与仿射群扩展和相应的表示理论相关的向量值设置中进行研究。这项工作将借鉴最近发展起来的一个理论，该理论将广义柯西-黎曼系统与包含仿射群的群的可单位表示联系起来。Torresani先前研究过的一个特别相关的例子是仿射群与海森堡群的半直接乘积。相应的帧分解将结合短时傅里叶变换和小波展开的各个方面。以小波和帧分解为重点的数学研究起源于经典的谐波分析，但最近受到信号处理研究的推动。在这里，小波概念被证明在分析由平稳和瞬态特征组成的信号方面非常有效。传统的傅立叶方法在这些研究中并不是有效的工具。在声学、数据压缩和偏微分方程研究中的应用已经证明了这些新发展思想的价值。＊＊＊

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. ***