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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Weyl-Heisenberg frames for subspaces of $L^2(R)$


Authors: Peter G. Casazza and Ole Christensen
Journal: Proc. Amer. Math. Soc. 129 (2001), 145-154
MSC (1991): Primary 42C15
Published electronically: July 27, 2000
MathSciNet review: 1784021
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Abstract: A Weyl-Heisenberg frame

\begin{displaymath}\{E_{mb}T_{na}g \}_{m,n \in Z} = \{ e^{2 \pi imb ( \cdot ) } g( \cdot - na) \}_{m,n \in Z}\end{displaymath}

for $L^2(R)$ allows every function $f \in L^2(R)$ to be written as an infinite linear combination of translated and modulated versions of the fixed function $g \in L^2(R)$. In the present paper we find sufficient conditions for $\{E_{mb}T_{na}g \}_{m,n \in Z}$ to be a frame for $\overline{span}\{E_{mb}T_{na}g \}_{m,n \in Z}$, which, in general, might just be a subspace of $L^2(R)$. Even our condition for $ \{E_{mb}T_{na}g \}_{m,n \in Z}$ to be a frame for $L^2(R)$ is significantly weaker than the previous known conditions. The results also shed new light on the classical results concerning frames for $L^2(R)$, showing for instance that the condition $G(x):= \sum_{n \in Z} \vert g(x-na)\vert^2 >A>0$is not necessary for $\{E_{mb}T_{na}g \}_{m,n \in Z}$ to be a frame for $\overline{span} \{E_{mb}T_{na}g \}_{m,n \in Z}$. Our work is inspired by a recent paper by Benedetto and Li, where the relationship between the zero-set of the function $G$ and frame properties of the set of functions $\{g( \cdot - n) \}_{n \in Z}$ is analyzed.


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Additional Information

Peter G. Casazza
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: pete@casazza.math.missouri.edu

Ole Christensen
Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark
Email: ole.christensen@mat.dtu.dk

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05731-2
PII: S 0002-9939(00)05731-2
Received by editor(s): March 10, 1999
Published electronically: July 27, 2000
Additional Notes: The first author was supported by NSF grant DMS 970618 and the second author by the Danish Research Council. The second author also thanks the University of Charlotte, NC, and the University of Missouri-Columbia, MO, for providing good working conditions.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society