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Geometric & Functional Analysis GAFA

, Volume 11, Issue 4, pp 742–758 | Cite as

Lattice tiling and the Weyl—Heisenberg frames

  • D. Han
  • Y. Wang

Abstract.

Let {\cal L} and {\cal K} be two full rank lattices in \( {\Bbb R}^d \). We prove that if \( {\rm v}({\cal L} ) = {\rm v}({\cal K}) \), i.e. they have the same volume, then there exists a measurable set \( \Omega \) such that it tiles \( {\Bbb R}^d \) by both \( {\cal L} \) and \( {\cal K} \). A counterexample shows that the above tiling result is false for three or more lattices. Furthermore, we prove that if \( {\rm v}({\cal L}) \le {\rm v}({\cal K}) \) then there exists a measurable set \( \Omega \) such that it tiles by \( {\cal L} \) and packs by \( {\cal K} \). Using these tiling results we answer a well-known question on the density property of Weyl—Heisenberg frames.

Keywords

Full Rank Density Property Tiling Result Lattice Tiling Rank Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel 2001

Authors and Affiliations

  • D. Han
    • 1
  • Y. Wang
    • 2
  1. 1.Department of Mathematics, University of Central Florida, Orlando, FL 32816-13364, USA, e-mail: dhan@pegasus.cc.ucf.eduUS
  2. 2.School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA, e-mail: wang@math.gatech.edu, http://www.math.gatech.edu/~wangUS

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