Zeros of the Zak transform on locally compact abelian groups
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- by Eberhard Kaniuth and Gitta Kutyniok
- Proc. Amer. Math. Soc. 126 (1998), 3561-3569
- DOI: https://doi.org/10.1090/S0002-9939-98-04450-5
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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.References
- Louis Auslander and Richard Tolimieri, Abelian harmonic analysis, theta functions and function algebras on a nilmanifold, Lecture Notes in Mathematics, Vol. 436, Springer-Verlag, Berlin-New York, 1975. MR 0414785, DOI 10.1007/BFb0069850
- John J. Benedetto and Michael W. Frazier (eds.), Wavelets: mathematics and applications, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1994. MR 1247511
- M. Boon and J. Zak, Amplitudes on von Neumann lattices, J. Math. Phys. 22 (1981), no. 5, 1090–1099. MR 622863, DOI 10.1063/1.524992
- Christopher E. Heil and David F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), no. 4, 628–666. MR 1025485, DOI 10.1137/1031129
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Die Grundlehren der mathematischen Wissenschaften, Band 115, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156915
- A. J. E. M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys. 23 (1982), no. 5, 720–731. MR 655886, DOI 10.1063/1.525426
- A. J. E. M. Janssen, The Zak transform: a signal transform for sampled time-continuous signals, Philips J. Res. 43 (1988), no. 1, 23–69. MR 947891
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- J. Zak, Finite translations in solid state physics, Phys. Rev. Letters 19 (1967), 1385-1387.
Bibliographic 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
- Received by editor(s): October 1, 1996
- Received by editor(s) in revised form: April 20, 1997
- Communicated by: J. Marshall Ash
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 3561-3569
- MSC (1991): Primary 43A32; Secondary 43A15, 43A40
- DOI: https://doi.org/10.1090/S0002-9939-98-04450-5
- MathSciNet review: 1459128