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A note on Gabor orthonormal bases

Author: Yun-Zhang Li
Journal: Proc. Amer. Math. Soc. 133 (2005), 2419-2428
MSC (2000): Primary 42C40
Published electronically: February 25, 2005
MathSciNet review: 2138885
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Abstract: The study of Gabor bases of the form $\{\, e^{-2\pi i\langle\lambda,\,\cdot\rangle}g(\cdot-m):\,\lambda,\, m\in {\mathbb Z}^n\,\}$ for $L^2({\mathbb R}^n)$ has interested many mathematicians in recent years. Alex Losevich and Steen Pedersen in 1998, Jeffery C. Lagarias, James A. Reeds and Yang Wang in 2000 independently proved that, for any fixed positive integer $n$, $\{\, e^{-2\pi i\langle\lambda,\,\cdot\rangle}:\,\lambda\in \Lambda\,\}$ is an orthonormal basis for $L^2([0,\,1]^n)$ if and only if $\{\, [0,\,1]^n+\lambda:\, \lambda\in \Lambda\,\}$ is a tiling of ${\mathbb R}^n$. Palle E. T. Jorgensen and Steen Pedersen in 1999 gave an explicit characterization of such $\Lambda$ for $n=1$, $2$, $3$. Inspired by their work, this paper addresses Gabor orthonormal bases of the form $\{\, e^{-2\pi i\langle\lambda,\,\cdot\rangle}g(\cdot-m):\,\lambda\in\Lambda,\, m\in {\mathbb Z}^n\,\}$ for $L^2({\mathbb R}^n)$ and some other related problems, where $\Lambda$ is as above. For a fixed $n\in \{\, 1,\, 2,\, 3\,\}$, the generating function $g$ of a Gabor orthonormal basis for $L^2({\mathbb R}^n)$ corresponding to the above $\Lambda$ is characterized explicitly provided that ${\mbox{supp}}(g)=[a_1,\,b_1]\times\cdots\times [a_n,\, b_n]$, which is new even if $\Lambda={\mathbb Z}^n$; a Shannon type sampling theorem about such $\Lambda$ is derived when $n=2$, $3$; for an arbitrary positive integer $n$, an explicit expression of the $g$with $\{\, e^{-2\pi i\langle\lambda,\,\cdot\rangle}g(\cdot-m):\,\lambda,\, m\in {\mathbb Z}^n\,\}$ being an orthonormal basis for $L^2({\mathbb R}^n)$ is obtained under the condition that $\vert\mbox{supp}(g)\vert=1$.

References [Enhancements On Off] (What's this?)

  • 1. Jeffery C. Lagarias, James A. Reeds and Yang Wang, Orthonormal Bases of Exponentials for the $n$-Cube, Duke Math. J., 103(2000), 25-37. MR 1758237 (2001h:11104)
  • 2. Palle E. T. Jorgensen and Steen Pedersen, Spectral Pairs in Cartesian Coordinates, J. Fourier Anal. Appl., 5(1999), 285-302. MR 1700084 (2002d:42027)
  • 3. O. H. Keller, Über die luckenlose Einfüllung des Raumes mit Würfeln, J. Reine und Angew. Math., 163(1930), 231-248.
  • 4. Jeffery C. Lagarias and P. Shor, Keller's Cube-Tiling Conjecture is False in High Dimensions, Bull. Amer. Math. Soc.(N. S.), 27(1992), 279-283. MR 1155280 (93e:52040)
  • 5. I. Daubechies, Ten Lectures on Wavelets, Philadelphia, 1992. MR 1162107 (93e:42045)
  • 6. A. Ron and Z. Shen, Weyl-Heisenberg Frames and Riesz Bases in $L^2({\mathbb R}^d)$, Duke Math. J., 89(1997), 237-282. MR 1460623 (98i:42013)
  • 7. Youming Liu and Yang Wang, The Uniformity of Non-uniform Gabor Bases, Adv. Comput. Math., 18(2003), 345-355. MR 1968125 (2004h:42036)
  • 8. O. Christensen, B. Deng and C. Heil, Density of Gabor frames, Appl. Comput. Harm. Anal., 7(1999), 292-304. MR 1721808 (2000j:42043)
  • 9. Deguang Han and Yang Wang, Lattice Tiling and the Weyl-Heisenberg Frames, Geom. Funct. Anal., 11(2001), 742-758. MR 1866800 (2003j:52021)
  • 10. M. A. Rieffel, Von Neumann Algebras associated with pairs of lattices in Lie Groups, Math. Ann., 257(1981), 403-413. MR 0639575 (84f:22010)
  • 11. J. Ramanathan and T. Steger, Incompleteness of Sparse Coherent States, Appl. Comput. Harm. Anal., 2(1995), 148-153. MR 1325536 (96b:81049)
  • 12. J. Benedetto, C. Heil and D. Walnut, Differentiation and the Balian-Low Theorem, J. Fourier Anal. Appl., 1(1995), 355-402. MR 1350699 (96f:42002)
  • 13. Peter G. Casazza and Ole Christensen, Gabor frames over irregular lattices. Frames, Adv. Comput. Math., 18(2003), 329-344. MR 1968124 (2004c:42062)
  • 14. Peter G. Casazza, Ole Christensen and A. J. E. M. Janssen, Weyl-Heisenberg frames, translation invariant systems and the Walnut representation,J. Funct. Anal., 180(2001),85-147. MR 1814424 (2002b:42042)
  • 15. J. Wexler and S. Raz, Discrete Gabor expansion, Signal Processing, 21(1990), 207-220.
  • 16. R. E. Edwards, Fourier series, A Modern Introduction, Volume 1, Second Edition, Springer-Verlag, 1979. MR 0545506 (80j:42001)
  • 17. K. Gröchenig and W. R. Madych, Multiresolution Analysis, Haar Bases, and Self-Similar Tilings of $\mathbb{R}^n$, IEEE Transactions on Information Theory, 38(1992), 556-568. MR 1162214 (93i:42001)
  • 18. Alex Iosevich and Steen Pedersen, Spectral and Tiling Properties of the Unit Cube, Internat. Math. Res. Notices, 1998, no. 16, 819-828. MR 1643694 (2000d:52015)
  • 19. X. Dai and D. R. Larson, Wandering Vectors for Unitary Systems and Orthogonal Wavelets, Mem. Amer. Math. Soc., 134(1998), no. 640. MR 1432142 (98m:47067)
  • 20. Yang Wang, Wavelets, Tiling, and Spectral Sets, Duke Math. J., 114(2002), 43-57. MR 1915035 (2003e:42057)

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Additional Information

Yun-Zhang Li
Affiliation: School of Applied Mathematics and Physics, Beijing University of Technology, Beijing, 100022, People’s Republic of China

Keywords: Gabor orthonormal basis, tiling
Received by editor(s): December 3, 2003
Received by editor(s) in revised form: April 19, 2004
Published electronically: February 25, 2005
Additional Notes: This research was supported by the National Natural Science Foundation of China, and the Natural Science Foundation of Beijing
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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