  We study the recovery of operators with bandlimited Kohn-Nirenberg symbol
from the action of such operators on a weighted impulse train, a procedure we
refer to as operator sampling. Kailath, and later Kozek and the authors have
shown that operator sampling is possible if the symbol of the operator is
bandlimited to a set with area less than one. In this paper we develop explicit
reconstruction formulas for operator sampling that generalize reconstruction
formulas for bandlimited functions. We give necessary and sufficient conditions
on the sampling rate that depend on size and geometry of the bandlimiting set.
Moreover, we show that under mild geometric conditions, classes of operators
bandlimited to an unknown set of area less than one-half permit sampling and
reconstruction. A similar result considering unknown sets of area less than one
was independently achieved by Heckel and Boelcskei.
  Operators with bandlimited symbols have been used to model doubly dispersive
communication channels with slowly-time-varying impulse response. The results
in this paper are rooted in work by Bello and Kailath in the 1960s.
