Continuous and discrete Gabor and wavelet analysis

本建议涉及现代傅立叶分析的不同领域，如小波理论，Gabor系统和时频分析。 研究工作的一个重要组成部分将涉及不规则Gabor和小波系统的理论。Gabor系统是用于“分析”给定信号的函数的集合。这个集合是由称为“窗口”的特定函数的移位和调制生成的。 如果这些调制和移位在时间-频率上规则地间隔开，则系统被称为“规则的”，如果它们不是，则系统被称为“不规则的”。虽然正规的伽柏系统的理论是很好的理解，目前，同样不能说不规则的系统。申请人建议更好地了解这些非正规系统，并改进他和其他人最近在这一领域所做的工作。 我们所提出的方法的主要优点之一是，而不是固定，先验，在时间-频率空间中的采样点的集合，然后试图找到可以有效地使用给定的采样集的窗口，我们固定的窗口（或双窗口对），然后试图找到一个合适的采样集在时间-频率空间。 小波系统也可以由一个单一的功能，但通过使用移位和伸缩，而不是，也可以定义一个规则和不规则小波系统的概念。 同样，在这一点上，规则小波系统的理论是相当发达的，而不规则小波系统仍处于起步阶段。我们建议调查申请人关于不规则Gabor系统的某些最近结果是否可以扩展到不规则小波系统。除了不规则Gabor和小波系统的理论之外，该建议还包括其他主题，例如子空间Gabor框架的所谓密度条件的研究和使用自仿射多瓦片构造的小波集的研究。

The present proposal involves different areas of modern Fourier analysis such as the theory of wavelets, Gabor systems and time -frequency analysis. One important component of the research effort will involve the theory of both irregular Gabor and wavelet systems. A Gabor systems is a collection of functions used to "analyse" a given signal. This collection is generated by shifts and modulations of a certain function called the "window". The system is called "regular" if these modulations and shifts are regularly spaced in the time-frequency and "irregular" if they are not. While the theory of regular Gabor systems is well understood at the present, the same cannot be said for irregular systems. The applicant proposes to develop a better understanding of these irregular systems and improves on recent work he and others have done in this area. One of the main advantages of our proposed approach to irregular Gabor systems is that, instead of fixing, a priori, the set of sampling points in the time-frequency space and then trying to find windows that can be use effectively with the given sampling set, we fix the window (or pair of dual windows) and then try to find a suitable sampling set in the time-frequency space. Wavelet systems can also be generated by a single function, but through the use of shifts and dilations instead, and a notion of regular and irregular wavelet system can also be defined. Again, the theory of regular wavelet systems is fairly well developed at this point while that of irregular wavelet system is still in its infancy. We propose to investigate whether certain recent results of the applicant concerning irregular Gabor systems can be extended to irregular wavelet systems. Beside the theory of irregular Gabor and wavelet systems, this proposal also includes other topics such as the study of so-called density conditions for subspace Gabor frames and the study of wavelet sets constructed using self-affine multi-tiles.