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Nonperiodic sampling and the local three squares theorem

Karlheinz Gröchenig Department of Mathematics, University of Connecticut Christopher Heil School of Mathematics, Georgia Institute of Technology David Walnut Department of Mathematical Sciences, George Mason University

TBD mathscidoc:1701.332931

Arkiv for Matematik, 38, (1), 77-92, 1998.9
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∈ L^{2}(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.
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@inproceedings{karlheinz1998nonperiodic,
  title={Nonperiodic sampling and the local three squares theorem},
  author={Karlheinz Gröchenig, Christopher Heil, and David Walnut},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203558101508740},
  booktitle={Arkiv for Matematik},
  volume={38},
  number={1},
  pages={77-92},
  year={1998},
}
Karlheinz Gröchenig, Christopher Heil, and David Walnut. Nonperiodic sampling and the local three squares theorem. 1998. Vol. 38. In Arkiv for Matematik. pp.77-92. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203558101508740.
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