Regular operator sampling for parallelograms

Operator sampling considers the question of when operators of a given class can be distinguished by their action on a single probing signal. The fundamental result in this theory shows that the answer depends on the area of the support S of the so-called spreading function of the operator (i.e., the symplectic Fourier transform of its Kohn-Nirenberg symbol). |S| <; 1 then identification is possible and when |S| > 1 it is impossible. In the critical case when |S| = 1, the picture is less clear. In this paper we characterize when regular operator sampling (that is, when the probing signal is a periodically-weighted delta train) is possible when S is a parallelogram of area 1.