Operator theory in function spaces on finitely-connected domains

我们提出研究在环（两个圆之间的区域）和壳（两个球之间的区域）等区域上定义的复函数空间。该项目有两个组成部分。首先，有一个非常理论化的部分，通过考虑作用于这些空间的某些变换，使我们能够更好地理解这些空间中的函数。其次，有一部分是由应用驱动的——通过近似技术，提出了一个框架，用于从区域边界上的不完整和噪声测量中恢复数据。虽然我们的贡献在很大程度上是理论上的，但它将构成数值算法的基础，这将对从事工程问题（如管道故障检测）和医学问题（特别是大脑扫描）的人感兴趣。

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).