Generalized sampling: extension to frames and inverse and ill-posed problems

Generalized sampling is a new framework for sampling and reconstruction in infinite-dimensional Hilbert spaces. Given measurements (inner products) of an element with respect to one basis, it allows one to reconstruct in another, arbitrary basis, in a way that is both convergent and numerically stable. However, generalized sampling is thus far only valid for sampling and reconstruction in systems that comprise bases. Thus, in the first part of this paper we extend this framework from bases to frames, and provide fundamental sampling theorems for this more general case. The second part of the paper is concerned with extending the idea of generalized sampling to the solution of inverse and ill-posed problems. In particular, we introduce two generalized sampling frameworks for such problems, based on regularized and non-regularized approaches. We furnish evidence of the usefulness of the proposed theories by providing a number of numerical experiments.

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