Exponential systems and related topics

这项研究项目是在调和分析领域，这是一个数学领域，它源于这样一个事实，即定义在一个区间上的许多函数可以分解为简单的正弦和余弦函数的和。首席调查员计划研究这样的分解不可能或效率不够高的情况，例如，因为所讨论的函数不再是按区间定义的。问题是，在这种情况下，是否可能进行类似的分解，将正弦和余弦替换为简单结构的其他函数。目标是使用以某种方式模拟正弦和余弦结构的函数作为三角函数的良好替代，三角函数为研究函数的性质以及它们之间的相互关系提供了一个很好的工具。特别是，这个项目包括开发有效的信号采样、内插和近似方法。主要研究人员和合作者最近证明，任何有限的区间并都允许指数的Riesz基。虽然这一领域取得了很大进展，但首席研究员仍将研究这个问题的许多方面，例如澄清这是否是所有有限测度集的通用性质，以及理解结果是否适用于具有“良好界”的Riesz基。此外，她计划为同样的目的研究使用更弱的概念(例如，帧、Riesz序列)的好处。特别令人感兴趣的是这种系统的研究与分析采样和内插序列的技术的发展之间的关系。最后，该项目将探索Gabor系统在时频分析中的应用，例如，巴利安-洛定理和Gabor元素的线性无关性的可能扩展。

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.