Nonperiodic sampling and the local three squares theorem

Karlheinz Gröchenig, Christopher Heil, David F. Walnut

Veröffentlichungen: Beitrag in FachzeitschriftArtikelPeer Reviewed


This paper presents an elementary proof of the following theorem: Given {rj}mj=1 with m=d+1, fix R=Smj=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.
Seiten (von - bis)77-92
FachzeitschriftArkiv för Matematik
PublikationsstatusVeröffentlicht - 2000

ÖFOS 2012

  • 1010 Mathematik


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