Operator theory in function spaces on finitely-connected domains

我们建议研究空间的复函数定义的区域，如环（两个圆之间的区域）和壳（两个球之间的区域）。该项目有两个组成部分。首先，有一个非常理论的部分，它使我们能够通过考虑作用在这些空间上的某些变换来更好地理解这些空间中的函数。第二，有一个部分是由应用程序的动机-通过近似技术，它被提议提供一个框架，用于从一个区域的边界上的不完整和嘈杂的测量恢复数据。虽然我们的贡献在很大程度上是理论性的，但它将成为数值算法的基础，这将是工程问题（如管道故障检测）和医学问题（特别是大脑扫描）的人们感兴趣的。

We propose to study spaces of complex functions defined on regions such as the annulus (a region between two circles) and the shell (a region between two spheres). The project has two components. First, there is a very theoretical part which enables us to understand better the functions in these spaces by considering certain transformations acting on them. Second, there is a part that is motivated by applications -- by means of an approximation technique, it is proposed to provide a framework for recovering data from incomplete and noisy measurements on the boundary of a region. Although our contribution is largely theoretical, it will form the basis of numerical algorithms, which will be of interest to people working in engineering problems (such as fault detection in pipes) and medical problems (in particular, brain scanning).