Continuous and discrete Gabor and wavelet analysis

目前的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系统和小波系统的理论。Gabor系统是用来“分析”给定信号的函数集合。这个集合是由一个叫做“窗口”的特定函数的移位和调制产生的。如果这些调制和移位在时间频率上有规则间隔，则系统称为“规则”，如果不是“不规则”。虽然规则Gabor系统的理论目前已被很好地理解，但对于不规则系统却不能说同样的话。申请人建议更好地了解这些不规范的系统，并改进他和其他人最近在这方面所做的工作。我们提出的不规则Gabor系统方法的主要优点之一是，我们不是先验地固定时频空间中的采样点集，然后尝试找到可以有效地使用给定采样集的窗口，而是固定窗口（或双窗口对），然后尝试在时频空间中找到合适的采样集。小波系统也可以由单个函数生成，但通过使用位移和膨胀代替，并且也可以定义规则和不规则小波系统的概念。同样，规则小波系统的理论在这一点上已经相当发达，而不规则小波系统的理论还处于起步阶段。我们打算研究申请者最近关于不规则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和小波系统的理论外，本提案还包括其他主题，如研究所谓的子空间Gabor框架的密度条件和研究使用自仿射多块构造的小波集。

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.