Nonperiodic sampling and the local three squares theorem
AbstractThis paper presents an elementary proof of the following theorem: Given {rj}jm=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 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. Whend=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.