Abstract
We present a simple proof of Ron and Shen's frame bounds estimates for Gabor frames. The proof is based on the Heil and Walnut's representation of the frame operator and shows that it can be decomposed into a continuous family of infinite matrices. The estimates then follow from a simple application of Gershgorin's theorem to each matrix. Next, we show that, if the window function has exponential decay, also the dual function has some exponential decay. Then, we describe a numerical method to compute the dual function and give an estimate of the error. Finally, we consider the spline of order 2; we investigate numerically the region of the time-frequency plane where it generates a frame and we compute the dual function for some values of the parameters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Daubechies, I. (1990). The wavelet transform, time-frequency localization and signal analysis,IEEE Trans. Inform. Theory,35, 961–1005.
Daubechies, I., Grossman, A., and Meyer, Y. (1988). Frames in the Bargmann space of entire functions,Comm. Pure Appl. Math.,XLI, 151–164.
Daubechies, I., Jaffard, S., and Journè, J.L. (1991). A simple Wilson orthonormal basis with exponential decay,SIAM J. Math. Anal,22, 554–572.
Daubechies, I., Landau, H.J., and Landau, Z. (1995). Gabor Time-frequency lattices and the Wexler-Raz identity,J. Fourier Anal. Appl.,I(4), 437–478.
Del Prete, V. (1997). Rational oversampling for Gabor frames,Technical report N. 353, University of Genoa, Italy.
Dixmier, V. (1969). Les Algèbres d'opérateurs dans l'espace Hilbertien, Gauthier-Villars, Paris.
Heil, C.E. and Walnut, D.F. (1989). Continuous and discrete wavelet transforms,SIAM Rev.,31(4), 628–666.
Janssen, A.J.E.M. (1995). On rationally oversampled Weyl-Heisenberg frames,IEEE Trans. Sign Proc.,47, 239–245.
Janssen, A.J.E.M. (1998). The duality condition for Weyl-Heisenberg frames, inGabor Analysis and Algorithms: Theory and Applications. Feichtinger, H.G. and Stromer, T., Eds., Birkhäuser, 33–84.
Janssen, A.J.E.M. (1995). Duality and biorthogonality for Weyl-Heisenberg frames,J. Four. Anal. Appl.,1, 404–437.
Janssen, A.J.E.M. (1997). From continuous to discrete Weyl-Heisenberg frames through sampling,J. Four. Anal Appl.,3, 583–595.
Janssen, A.J.E.M. (1996). Some Weyl-Heisenberg frame bound calculations,Indag. Mathem.,7, 165–182.
Lyubarskii, Y. (1992). Frames in the Bargmann space of entire functions, in Entire and Subharmonic functions. Ya Levin, B., Ed.,Advances in Soviet Mathematics,11, Springer-Verlag, Berlin.
Qiu, S. and Feichtinger, H.G. (1995). Discrete Gabor structures and optimal representation,IEEE Trans. Sign. Proc.,43, 2258–2268.
Ron, A. and Shen, Z. (1997). Weyl-Heisenberg frames and Riesz bases inL 2(Rd),Duke Math. J.,89, 237–282.
Seip, K. and Wallsten, R. (1992). Density Theorems for sampling and interpolation in the Bargmann-Fock space, II,J. Reine Angew. Math.,429, 107–113.
Tolimieri, R. and Orr, S. (1995). Poisson Summation, the ambiguity function and the theory of Weyl-Heisenberg frames,J. Four. Anal. Appl.,1, 233–247.
Wexler, J. and Raz, S. (1990). Discrete Gabor expansion,Signal Proc.,21, 207–220.
Zibulski, M. and Zeevi, Y.Y. (1993). Oversampling in the Gabor scheme,IEEE Trans. Sign. Proc.,41(8), 2679–2687.
Zibulski, M. and Zeevi, Y.Y. (1997). Analysis of multiwindow Gabor-type schemes by Frame Methods,Appl. Comput. Harmonic Anal.,4, 188–221.
Author information
Authors and Affiliations
Additional information
Communicated by Hans Feichtinger and Guido Weiss
Rights and permissions
About this article
Cite this article
Del Prete, V. Estimates, decay properties, and computation of the dual function for Gabor frames. The Journal of Fourier Analysis and Applications 5, 545–562 (1999). https://doi.org/10.1007/BF01257190
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01257190