CAREER: Bases in Hilbert Function Spaces and Some of their Applications

这个项目讨论了不同的方法，通过这些方法可以将函数分解或近似为具有简单结构的函数之和。特别感兴趣的是分解成简单的正弦和余弦函数的总和。 对于定义在区间上的函数，或者定义在立方体上的高维函数，这样的分解是经典的研究领域。这个项目考虑在更复杂的集合上定义的函数，并研究在这种设置中存在类似的分解。以这种方式近似或分解函数的能力为研究提供了强大的工具，可以应用于各种环境。这个理论的几个应用将在整个项目中进行研究。作为该项目的一部分，PI将开发，组织和教授一个名为“亚特兰大地区学生分析荣誉课程”的新课程。这个为期三年的计划，为有前途的本科生，预计将增加其参与者的动机和潜力被录取到领先的研究生课程。众所周知，大多数可测集不允许指数或三角函数的正交基。因此，要在这样的集合上使用调和分析机器，标准正交基的刚性结构需要被更灵活的Riesz基结构所取代，或者如果这是不可能的，那么可以用它的放松版本，如框架和Riesz序列。在这个项目中，PI将继续（与合作者）研究这些问题。例如，她将分析关于指数的Riesz基的存在性问题，以及与具有普适性质的系统相关的问题。此外，PI将把这一理论应用于几种不同环境下的数学问题，包括带限信号的采样和插值，某些不确定性原理的扩展和改进，以及高斯噪声在长时间间隔内的行为。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project discusses different ways by which a function can be decomposed, or approximated, by sums of functions with a simple structure. Of particular interest are decompositions into sums of the simple sine and cosine functions. For functions defined over an interval, or in higher dimensions over a cube, such decompositions are a classical area of study. This project considers functions defined over more complicated sets, and studies the existence of similar decompositions in such settings. The ability to approximate or decompose functions in such a way provides a strong tool for research, which can be applied in various settings. Several applications of this theory will be studied throughout this project. As part of this project, the PI will develop, organize and teach in a new program titled "Honors level Program in Analysis for Students in the Atlanta Area". This three-year program, for promising undergraduate students, is expected to increase both the motivation and the potential of its participants to be admitted to leading graduate programs. It is well known that most measurable sets do not admit an orthogonal basis of exponentials, or of trigonometric functions. Therefore, to use harmonic analysis machinery over such sets, the rigid structure of an orthonormal basis needs to be replaced by the more flexible structure of a Riesz basis, or if this is impossible, by relaxed versions of it such as frames and Riesz sequences. In this project, the PI will continue her study (with collaborators) of such problems. For example, she will analyze questions regarding the existence of Riesz bases of exponentials, and problems related to systems with universal properties. Further, the PI will apply this theory to mathematical problems in several different settings, including sampling and interpolation of band limited signals, extensions and refinements of certain uncertainty principles, and the behavior of Gaussian noise over long intervals of time.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.