Operator theory in function spaces on finitely-connected domains

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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).