Multidimensional Wavelets in Non-Isotropic Function Spaces

本计画探讨调和分析与小波领域中有关多维小波展开之数学理论。该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，小波具有良好的时频特性。这些领域的研究由于I.Daubechies，R. Coifman和Y. Meyer，沿着还有许多其他的研究。然而，大部分的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理论的问题没有回答。提出者将从三个方向对非各向同性小波理论进行研究。第一部分是研究标准函数空间的非各向同性类似的可扩张扩张空间，特别是研究非各向同性扩张的Calderon-Zygmund算子的小波展开特征。第二是对大类非各向同性膨胀膨胀函数构造具有良好时频局部化特性的正交小波。第三是识别非各向同性膨胀膨胀，对于这些膨胀，构造良好的局部化小波是不可能的。更一般地说，这个建议代表了小波分析的工作，小波分析是调和分析中的一种强有力的技术。这种技术在信号和图像处理中产生了广泛的应用，例如JPEG 2000图像压缩系统。可以预期，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。

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.