Nonperiodic sampling and the local three squares theorem

Abstract

This paper presents an elementary proof of the following theorem: Given {r_j} ${}_{j}^{m}$ =1 with m=d+1, fix $fix R \geqslant \sum\nolimits_{j = 1}^m {r_j } $ and let Q=[−R, R]^d. Then any f∈ L²(Q) is completely determined by its averages over cubes of side r_j that are completely contained in Q and have edges parallel to the coordinate axes if and only if r_j/r_k 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.