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

 

Zeros of the Zak transform
on locally compact Abelian groups


Authors: Eberhard Kaniuth and Gitta Kutyniok
Journal: Proc. Amer. Math. Soc. 126 (1998), 3561-3569
MSC (1991): Primary 43A32; Secondary 43A15, 43A40
MathSciNet review: 1459128
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Abstract: Let $G$ be a locally compact abelian group. The notion of Zak transform on $L^2(\mathbb{R}^d)$ extends to $L^2(G)$. Suppose that $G$ is compactly generated and its connected component of the identity is non-compact. Generalizing a classical result for $L^2(\mathbb{R})$, we then prove that if $f \in L^2(G)$ is such that its Zak transform $Z f$ is continuous on $G \times \widehat{G}$, then $Z f$ has a zero.


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Additional Information

Eberhard Kaniuth
Affiliation: Fachbereich Mathematik/Informatik, Universität Paderborn, 33095 Paderborn, Germany
Email: kaniuth@uni-paderborn.de

Gitta Kutyniok
Affiliation: Fachbereich Mathematik/Informatik, Universität Paderborn, 33095 Paderborn, Germany
Email: gittak@uni-paderborn.de

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04450-5
PII: S 0002-9939(98)04450-5
Received by editor(s): October 1, 1996
Received by editor(s) in revised form: April 20, 1997
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society