The classical sampling theorem, attributed to Whittaker, Shannon, Nyquist, and Kotelnikov, states that a
bandlimited function can be recovered from its samples, as long as we use a sufficiently dense sampling grid.
Here, we review the recent development of an operator sampling theory which allows for a "widening" of the
classical sampling theorem. In this realm, bandlimited functions are replaced by "bandlimited operators". that
is, by pseudodifferential operators which have bandlimited
Similar to the Nyquist sampling density condition alluded to above, we discuss sufficient and necessary
conditions on the bandlimitation of pseudodifferential operators to ensure that they can be recovered by their
action on a single distribution. In fact, we show that an operator with Kohn-Nirenberg symbol bandlimited to
a Jordan domain of measure less than one can be recovered through its action on a distribution defined on a
appropriately chosen sampling grid. Further, an operator with bandlimitation to a Jordan domain of measure
larger than one cannot be recovered through its action on any tempered distribution whatsoever, pointing towards
a fundamental difference to the classical sampling theorem where a large bandwidth could always be compensated
through a sufficiently fine sampling grid. The dichotomy depending on the size of the bandlimitation is related
to Heisenberg's uncertainty principle.
Further, we discuss an application of this theory to the channel measurement problem for Multiple-Input
Multiple-Output (MIMO) channels.