Continuous and discrete Gabor and wavelet analysis

目前的建议涉及现代傅里叶分析的不同领域，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系统和小波系统的理论。Gabor系统是用于“分析”给定信号的函数的集合。这个集合是通过对被称为“窗口”的特定函数的移位和调制来产生的。如果这些调制和移位在时间-频率上按规则间隔，则该系统被称为“规则”，如果它们不是按规则间隔的，则称为“不规则”。虽然正则Gabor系统的理论目前已经很好地被理解了，但对于非规则系统却不能说同样的话。申请人提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。对于非规则Gabor系统，我们提出的方法的主要优点之一是，不是先确定时频空间中的采样点集合，然后尝试寻找能够有效利用给定采样集的窗口，而是固定窗口(或对偶窗口)，然后尝试在时频空间中找到合适的采样集。小波系也可以由单个函数产生，但通过使用移位和伸缩来产生，并且还可以定义规则和不规则小波系的概念。再次，在这一点上，正则小波系统的理论已经相当发达，而非规则小波系统的理论还处于初级阶段。我们建议调查申请人最近关于不规则Gabor系统的某些结果是否可以推广到不规则小波系统。除了不规则Gabor系统和小波系统的理论外，这一建议还包括对子空间Gabor框架的所谓密度条件的研究和对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.