  In this paper, we introduce a computational framework for recovering a
high-resolution approximation of an unknown function from its low-resolution
indirect measurements as well as high-resolution training observations by
merging the frameworks of generalized sampling and functional principal
component analysis. In particular, we increase the signal resolution via a data
driven approach, which models the function of interest as a realization of a
random field and leverages a training set of observations generated via the
same underlying random process. We study the performance of the resulting
estimation procedure and show that high-resolution recovery is indeed possible
provided appropriate low-rank and angle conditions hold and provided the
training set is sufficiently large relative to the desired resolution.
Moreover, we show that the size of the training set can be reduced by
leveraging sparse representations of the functional principal components.
Furthermore, the effectiveness of the proposed reconstruction procedure is
illustrated by various numerical examples.
