Generalised sampling in Hilbert spaces
Ben Adcock, (Department of Mathematics, Simon Fraser University, Canada)

Seminar • 20 June 2011 • MEB 10, EPFL

Abstract
The purpose of this talk is to present a new framework for the problem of reconstructing an object - a signal or image, for example - in an arbitrary basis from its measurements with respect to certain sampling vectors. Unlike more conventional approaches, such as consistent reconstructions, this method is both numerically stable and guaranteed to converge as the number of samples increases. Moreover, the accuracy of the reconstruction is determined solely by the reconstruction space istelf, and not by the nature of the sampling. The key ingredient in this framework is oversampling. By allowing the number of samples to be greater than the number of degrees of freedom in the reconstruction, one obtains both numerical stability and convergence. In addition, the amount of oversampling required can be determined by a quantity known as the stable sampling rate. This quantity is easily computable, and therefore stability and convergence can both be guaranteed a priori. The ideas for generalised sampling stem from considerations about how to discretise certain infinite-dimensional operators. In the final part of this talk I shall describe this matter in more detail, and briefly discuss applications to several related problems, including so-called infinite-dimensional compressed sensing. This is joint work with Anders Hansen (DAMTP, University of Cambridge)