Nonperiodic sampling and reconstruction from averages

We show that recovery of a function from its averages over squares in the plane is closely related to a problem of recovery of bandlimited functions from samples on unions of regular lattices. We use this observation to construct explicit solutions to the Bezout equation which can be easily implemented in software. We also show that these sampling results give a new proof of the “three squares theorem” which says that a function in the plane can be recovered from its averages on translates of three squares oriented parallel to the coordinate axes whose sidelengths are pairwise irrationally related. Other proofs of this theorem and construction of solution to the Bezout equation rely on interpolation methods in the theory of functions of one and several complex variables. Our sampling technique gives much simpler solutions especially in higher dimensions