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The Uniformity of Non-Uniform Gabor Bases

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Abstract

There have been extensive studies on non-uniform Gabor bases and frames in recent years. But interestingly there have not been a single example of a compactly supported orthonormal Gabor basis in which either the frequency set or the translation set is non-uniform. Nor has there been an example in which the modulus of the generating function is not a characteristic function of a set. In this paper, we prove that in the one dimension and if we assume that the generating function g(x) of an orthonormal Gabor basis is supported on an interval, then both the frequency and the translation sets of the Gabor basis must be lattices. In fact, the Gabor basis must be the “trivial” one in the sense that |g(x)|=cχΩ(x) for some fundamental interval of the translation set. We also give examples showing that compactly supported non-uniform orthonormal Gabor bases exist in higher dimensions.

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References

  1. J.J. Benedetto, C. Heil and D.F. Walnut, Gabor systems and the Balian-Low theorem, in: Gabor Analysis and Algorithms, Applied and Numerical Harmonic Analysis (Birkhäuser, Boston, MA, 1998) pp. 85–122.

    Google Scholar 

  2. A. Beurling and P. Malliavin, On the closure of a sequence of exponentials on a segment, Multigraphed lectures, Summer Institute on Harmonic Analysis, Stanford University, Stanford, CA (1961).

    Google Scholar 

  3. R. Boas, Entire Functions (Academic Press, New York, 1954).

    Google Scholar 

  4. P. Casazza, Modern tools forWeyl-Heisenberg (Gabor) frame theory, Adv. in Imaging Electron. Phys. 115 (2000) 1–127.

    Google Scholar 

  5. P. Casazza and O. Christensen, Classifying certain irregular Gabor frames, Preprint.

  6. J.W.S. Cassels, An Introduction to Diophantine Approximation, Cambridge Tracts, No. 45 (Cambridge Univ. Press, London, 1957).

    Google Scholar 

  7. O. Christensen, B. Deng and C. Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal. 7 (1999) 292–304.

    Google Scholar 

  8. I. Daubechies, H. Landau and Z. Landau, Gabor time frequency lattices and the Wexler-Raz identity, J. Fourier. Anal. Appl. 1 (1995) 437–478.

    Google Scholar 

  9. R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366.

    Google Scholar 

  10. H.G. Feichtinger and T. Strohmer, eds., Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis (Birkhäuser, Basel, 1998).

    Google Scholar 

  11. B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem, J. Funct. Anal. 16 (1974) 101–121.

    Google Scholar 

  12. D. Gabor, Theory of communication, J. Inst. Elec. Engrg. (London) 93 (1946) 429–457.

    Google Scholar 

  13. K. Gröchenig, Foundations of Time-Frequency Analysis (Birkhäuser, Basel, 2000).

    Google Scholar 

  14. K. Gröchenig and H. Razafinjatovo, On Landau's necessary density conditions for sampling and interpolation of band-limited functions, J. London Math. Soc. (2) 54 (1996) 557–565.

    Google Scholar 

  15. D. Han and Y. Wang, Lattice tilings and Weyl-Heisenberg frames, Geom. Funct. Anal. 11 (2001) 742–758.

    Google Scholar 

  16. A. Iosevich and S. Pedersen, Spectral and tiling properties of the unit cube, Internat. Math. Res. Notices 16 (1998) 819–828.

    Google Scholar 

  17. P.E.T. Jorgensen and S. Pedersen, Spectral pairs in cartesian coordinates, J. Fourier Anal. Appl. 5 (1999) 285–302.

    Google Scholar 

  18. M. Koloutzakis and J. Lagarias, Structure of tilings of the line by a function, Duke Math. J. 82 (1996) 653–678.

    Google Scholar 

  19. I. Laba, Fuglede's conjecture for a union of two intervals, Proc. Amer. Math. Soc. 129 (2001) 2965–2972.

    Google Scholar 

  20. I. Laba and Y. Wang, Spectral measures, Preprint.

  21. J.C. Lagarias, J.A. Reed and Y. Wang, Orthonormal bases of exponentials for the n-cube, Duke Math. J. 103 (2000) 25–37.

    Google Scholar 

  22. J. Lagarias and P. Shor, Keller's cube-tiling conjecture is false in high dimensions, Bull. Amer. Math. Soc. 27 (1992) 279–283.

    Google Scholar 

  23. J.C. Lagarias and Y. Wang, Tiling the line by the translates of one tile, Invent. Math. 124 (1996) 341–365.

    Google Scholar 

  24. J.C. Lagarias and Y. Wang, Spectral sets and factorizations of finite Abelian groups, J. Funct. Anal. 145 (1997) 73–98.

    Google Scholar 

  25. H. Landau, Necessary denisty conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967) 37–52.

    Google Scholar 

  26. J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2 (1995) 148–153.

    Google Scholar 

  27. M.A. Rieffel, Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981) 403–413.

    Google Scholar 

  28. A. Ron and Z. Shen, Weyl-Heisenberg frames and Riesz bases in L 2 (d ), Duke Math. J. 89 (1997) 237–282.

    Google Scholar 

  29. K. Seip, On the connection between exponential bases and certain related sequences in L 2 (, π), J. Funct. Anal. 130 (1995) 131–160.

    Google Scholar 

  30. R. Young, An Introduction to Nonharmonic Fourier Series (Academic Press, New York, 1980).

    Google Scholar 

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Liu, Y., Wang, Y. The Uniformity of Non-Uniform Gabor Bases. Advances in Computational Mathematics 18, 345–355 (2003). https://doi.org/10.1023/A:1021350103925

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