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Remarks on multivariate Gaussian Gabor frames

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Abstract

Spanning properties of multivariate Gaussian Gabor systems are far from being fully understood. Our results illustrate that, unlike in dimension one where Gaussian Gabor frames are characterized in terms of lattice density, the behavior of Gaussian Gabor systems in higher-dimensions is intricate and further exploration is a valuable and challenging task.

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Notes

  1. Note that the spanning properties of \((\chi _{[0,1)},a\mathbb{Z }\times b\mathbb{Z })\) depend in a quite delicate matter on the choice of the positive parameters \(a\) and \(b\) [16].

  2. The equality \(\displaystyle V_\mathfrak{g _d}f(x,-\xi )=e^{2\pi ix\xi }\mathfrak B f(x+i\xi )e^{-\frac{\pi }{2}|x+iy|^2}\), \(f\in L^2(\mathbb{R }^d)\), where \(\mathfrak B \) denotes the Bargmann transform, describes the correspondence between the study of Gaussian Gabor frames and sampling in Bargmann–Fock space.

  3. For details on the use of the Zak transform in Gabor analysis, see Section 9.4 in [1, 5] for the one-dimensional case and Chapter 8 in [10] for the multi-dimensional case.

References

  1. Bastiaans, M.J.: Gabor’s expansion and the Zak transform for continuous-time and discrete-time signals: critical sampling and rational oversampling. Eindhoven University of Technology research reports. Eindhoven University of Technology, Eindhoven (1995)

  2. Bekka, B.: Square integrable representations, von Neumann algebras and an application to Gabor analysis. J. Fourier Anal. Appl. 10(4), 325–349 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benedetto, J.J., Heil, C., Walnut, D.: Gabor analysis and the Balian–Low theorem. In: Feichtinger, H.G., Strohmer, T. (eds.) Gabor Analysis and Algorithms, chapter 2, pp. 84–122. Birkhäuser, Boston (1998)

  4. Bourouihiya, A.: The tensor product of frames. Sampl. Theory Signal Image Process. 7(1), 65–76 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Christensen, O.: An introduction to frames and Riesz bases. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston (2003)

    Google Scholar 

  6. Daubechies, I.: The wavelet transform, time-frequency localization and signal analysis. IEEE Trans. Inform. Theory 36(5), 961–1005 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. de Gosson, M: Construction of multivariate Gaussian Weyl-Heisenberg frames (1). Preprint

  8. Folland, G.B.: Harmonic analysis in phase space. Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)

    Google Scholar 

  9. Gabor, D.: Theory of communication. J. Inst. Elect. Eng. (London) 93(3), 429–457 (1946)

    Google Scholar 

  10. Gröchenig, K.: Foundations of time-frequency analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston (2001)

    Google Scholar 

  11. Gröchenig, K.: Multivariate Gabor frames and sampling of entire functions of several variables. Appl. Comput. Harmon. A 31(2), 218–227 (2011)

    Article  MATH  Google Scholar 

  12. Gröchenig, K., Han, D., Heil, C., Kutyniok, G.: The Balian–Low theorem for symplectic lattices in higher dimensions. Appl. Comput. Harmon. A 13(2), 169–176 (2003)

    Article  Google Scholar 

  13. Heil, C.: History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl. 13(2), 113–166 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Janssen, A.J.E.M.: Gabor representation of generalized functions. J. Math. Anal. Appl. 83(2), 377–394 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Janssen, A.J.E.M.: Signal analytic proof of two basic results on lattice expansions. Appl. Comput. Harmon. A 1(4), 350–354 (1994)

    Article  MATH  Google Scholar 

  16. Janssen, A. J. E. M.: Zak transforms with few zeros and the tie. In: Advances in Gabor analysis (Appl. Numer. Harmon. Anal.) pp. 31–70. Birkhäuser Boston, Boston (2003)

  17. Kutyniok, G.: Beurling density and shift-invariant weighted irregular Gabor systems. Sampl. Theory Signal Image Process. 5(2), 163–181 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Lyubarskiĭ, Y.I.: Frames in the Bargmann space of entire functions. In: Entire and subharmonic functions, advances in Soviet mathematics, vol. 11, pp. 167–180. American Mathematical Society, Providence (1992)

  19. Ortega-Cerdà, J., Schuster, A., Varolin, D.: Interpolation and sampling hypersurfaces for the Bargmann–Fock space in higher dimensions. Math. Ann. 335(1), 79–107 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pfander, G.E., Rashkov, P.: Window design for multivariate Gabor frames on lattices. Technical report 21. Jacobs University, Bremen (2010)

  21. Seip, K., Wallstén, R.: Density theorems for sampling and interpolation in the Bargmann–Fock space. II. J. Reine Angew. Math. 429, 107–113 (1992)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Engineering and Science, Jacobs University, 28759 , Bremen, Germany

    Götz E. Pfander

  2. Fachbereich Mathematik und Informatik, Philipps Universität Marburg, 35032 , Marburg, Germany

    Peter Rashkov

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  1. Götz E. Pfander
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  2. Peter Rashkov
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Correspondence to Götz E. Pfander.

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Communicated by K. Gröchenig.

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Pfander, G.E., Rashkov, P. Remarks on multivariate Gaussian Gabor frames. Monatsh Math 172, 179–187 (2013). https://doi.org/10.1007/s00605-013-0556-4

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