Journal of Fourier Analysis and Applications

, Volume 22, Issue 6, pp 1440–1451 | Cite as

Counterexamples to the B-spline Conjecture for Gabor Frames

Article

Abstract

The frame set conjecture for B-splines \(B_n\), \(n \ge 2\), states that the frame set is the maximal set that avoids the known obstructions. We show that any hyperbola of the form \(ab=r\), where r is a rational number smaller than one and a and b denote the sampling and modulation rates, respectively, has infinitely many pieces, located around \(b=2,3,\dots \), not belonging to the frame set of the nth order B-spline. This, in turn, disproves the frame set conjecture for B-splines. On the other hand, we uncover a new region belonging to the frame set for B-splines \(B_n\), \(n \ge 2\).

Keywords

B-spline Frame Frame set Gabor system Zibulski–Zeevi matrix 

Mathematics Subject Classification

Primary 42C15 Secondary 42A60 

Notes

Acknowledgments

The authors would like to thank Ole Christensen for posing the problem of characterizing the frame set of B-splines in a talk at the Technical University of Denmark on February 3, 2015, that initiated the work presented in this paper. The authors would also like to thank Karlheinz Gröchenig and the referees for comments improving the presentation of the paper.

References

  1. 1.
    Aldroubi, A., Gröchenig, K.: Beurling-Landau-type theorems for non-uniform sampling in shift invariant spline spaces. J. Fourier Anal. Appl. 6(1), 93–103 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Christensen, O.: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston (2003)CrossRefMATHGoogle Scholar
  3. 3.
    Christensen, O.: Six (seven) problems in frame theory. In: Zayed, A.I., Schmeisser, G. (eds.) New Perspectives on Approximation and Sampling Theory, Applied and Numerical Harmonic Analysis, pp. 337–358. Springer, New York (2014)Google Scholar
  4. 4.
    Christensen, O., Kim, H.O., Kim, R.Y.: On Gabor frames generated by sign-changing windows and B-splines. Appl. Comput. Harmon. Anal. 39(3), 534–544 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dai, X.-R., Sun, Q.: The abc problem for Gabor systems. Memoirs of American Mathematical Society, to appear, p 106 (2014)Google Scholar
  6. 6.
    Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27(5), 1271–1283 (1986)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Del Prete, V.: Estimates, decay properties, and computation of the dual function for Gabor frames. J. Fourier Anal. Appl. 5(6), 545–562 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Del Prete, V.: On a necessary condition for B-spline Gabor frames. Ric. Mat. 59(1), 161–164 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Feichtinger, H.G., Kaiblinger, N.: Varying the time-frequency lattice of Gabor frames. Trans. Am. Math. Soc. 356(5), 2001–2023 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gröchenig, K.: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser Boston Inc., Boston (2001)CrossRefMATHGoogle Scholar
  11. 11.
    Gröchenig, K.: Partitions of unity and new obstructions for gabor frames. arXiv:1507.08432
  12. 12.
    Gröchenig, K.: The mystery of Gabor frames. J. Fourier Anal. Appl. 20(4), 865–895 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gröchenig, K., Janssen, A.J.E.M., Kaiblinger, N., Pfander, G.E.: Note on \(B\)-splines, wavelet scaling functions, and Gabor frames. IEEE Trans. Inf. Theory 49(12), 3318–3320 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jakobsen, M.S., Lemvig, J.: Co-compact Gabor systems on locally compact abelian groups. J. Fourier Anal. Appl. 22(1), 36–70 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    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)Google Scholar
  16. 16.
    Kloos, T., Stöckler, J.: Zak transforms and Gabor frames of totally positive functions and exponential B-splines. J. Approx. Theory 184, 209–237 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Lyubarskii, Y., Nes, P.G.: Gabor frames with rational density. Appl. Comput. Harmon. Anal. 34(3), 488–494 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark

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