Exponential systems and related topics

这个研究项目是在谐波分析领域，这是一个数学领域，它源于这样一个事实，即在一个区间内定义的许多函数可以分解为简单的正弦和余弦函数的和。主要研究者计划研究这样的分解是不可能的，或者是不够有效的情况，例如，因为所讨论的函数不再在一段时间内定义。问题是，在这种情况下，用简单结构的其他函数代替正弦和余弦函数，是否有可能进行类似的分解。我们的目标是用函数来模拟正弦和余弦函数的结构，以这样或那样的方式，作为三角函数的很好的替代品，三角函数为研究函数的性质和它们之间的相互关系提供了一个很好的工具。特别地，这个项目包括开发采样、插值和近似信号的有效方法。首席研究员和合作者最近证明，任何区间的有限并都承认指数的Riesz基。虽然在该领域取得了很大的进步，但仍有许多方面的问题需要首席研究员进行研究，例如澄清这是否是所有有限测度集的一般性质，以及理解结果是否适用于具有“良好边界”的Riesz基。此外，她计划研究使用更弱的概念（例如，框架，Riesz序列）的好处为相同的目的。特别令人感兴趣的是这种系统的研究与分析采样和插值序列的技术发展之间的关系。最后，该项目将探索Gabor系统在时频分析中的应用，例如，bali - low定理的可能扩展和Gabor元素的线性独立性。

This research project is in the field of harmonic analysis, a mathematical area that grew from the fact that many functions defined over an interval can be decomposed as sums of the simple sine and cosine functions. The principal investigator plans to study cases where such a decomposition is not possible, or is not efficient enough, for example because the functions in question are no longer defined over intervals. The question is whether similar decompositions are possible in such cases, with the sines and cosines replaced by other functions of a simple structure. The goal is to use functions that mimic the structure of the sines and cosines, in one way or another, as good replacements for the trigonometric functions, which provide an excellent tool for studying properties of functions and the interrelationships between them. In particular, this project includes the development of efficient ways to sample, interpolate, and approximate signals.The principal investigator and collaborators have recently shown that any finite union of intervals admits a Riesz basis of exponentials. While a big advance in the area, there remain many aspects of the question that the principal investigator will study, such as clarifying whether this is a generic property of all sets of finite measure and understanding whether the result holds for Riesz bases with "good bounds." Further, she plans to study the benefits of using even weaker notions (e.g., frames, Riesz sequences) for the same purposes. Of particular interest is the relation between the study of such systems and the development of techniques to analyze sampling and interpolating sequences. Finally, the project will explore applications to Gabor systems in time-frequency analysis, for instance, possible extensions of the Balian-Low theorem and linear independence of Gabor elements.