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The geometry of sampling on unions of lattices

Author: Eric Weber
Journal: Proc. Amer. Math. Soc. 132 (2004), 3661-3670
MSC (2000): Primary 42B05; Secondary 94A20
Published electronically: June 21, 2004
MathSciNet review: 2084089
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Abstract: In this paper we show two results concerning sampling translation-invariant subspaces of $L^2({\mathbb R}^d)$ on unions of lattices. The first result shows that the sampling transform on a union of lattices is a constant times an isometry if and only if the sampling transform on each individual lattice is so. The second result demonstrates that the sampling transforms of two unions of lattices on two bands have orthogonal ranges if and only if, correspondingly, the sampling transforms of each pair of lattices have orthogonal ranges. We then consider sampling on shifted lattices.

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Additional Information

Eric Weber
Affiliation: Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071-3036
Address at time of publication: Department of Mathematics, 400 Carver Hall, Iowa State University, Ames, Iowa 50011

Received by editor(s): November 4, 2002
Received by editor(s) in revised form: August 26, 2003
Published electronically: June 21, 2004
Additional Notes: This research was supported in part by NSF grants DMS-0200756 and DMS-0308634.
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society