Nonperiodic sampling and the local three squares theorem

Abstract

This paper presents an elementary proof of the following theorem: Given {rj}mj=1 with m=d+1, fix R≥Σmj=1rj and let Q=[-R, R]d. Then any f ∈ L2(Q) is completely determined by its averages over cubes of side rj that are completely contained in Q and have edges parallel to the coordinate axes if and only if rj/rk is irrational for j≠k. When d=2 this theorem is known as the local three squares theorem and is an example of a Pompeiu-type theorem. The proof of the theorem combines ideas in multisensor deconvolution and the theory of sampling on unions of rectangular lattices having incommensurate densities with a theorem of Young on sequences biorthogonal to exact sequences of exponentials.