Multidimensional wavelets in non-isotropic function spaces

该项目涉及有关多维小波展开数学理论的调和分析方面的教育和研究活动。如何构造具有良好时频局部化等性质的小波是本课题的基本问题之一。尽管在这一领域取得了长足的进展，但大多数研究都集中在各向同性理论上，而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发展构造非各向同性膨胀大类的局部化良好的正交小波的技术。一个密切相关的互补主题是识别不可能构造良好局部化小波的非各向同性伸缩。该项目的另一个方向是研究与伸缩相关的经典函数空间的非各向同性类似。更广泛地说，该项目代表了小波分析方面的工作，这是一种强大的谐波分析技术。与各向同性小波理论相比，高维各向同性小波理论在纯分析和信号与图像处理的更多应用领域都具有广阔的应用前景。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。

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.