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A necessary and sufficient condition for dual Weyl-Heisenberg frames to be compactly supported. (English) Zbl 0933.42029

Summary: We consider continuous-time Weyl-Heisenberg (Gabor) frame expansions with rational oversampling. We present a necessary and sufficient condition on a compactly supported function \(g(t)\) generating a Weyl-Heisenberg frame for \(L^2(\mathbb{R})\) for its minimal dual (Wexler-Raz dual) \(\gamma^0(t)\) to be compactly supported. We furthermore provide a necessary and sufficient condition for a band-limited function \(g(t)\) generating a Weyl-Heisenberg frame for \(L^2(\mathbb{R})\) to have a band-limited minimal dual \(\gamma^0(t)\). As a consequence of these conditions, we show that in the cases of integer oversampling and critical sampling a compactly supported (band-limited) \(g(t)\) has a compactly supported (band-limited) minimal dual \(\gamma^0(t)\) if and only if the Weyl-Heisenberg frame operator is a multiplication operator in the time (frequency) domain. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Weyl-Heisenberg frame operator, and on the theory of polynomial matrices.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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References:

[1] Bastiaans, M.J. (1980). Gabor’s expansion of a signal in Gaussian elementary signals,Proc. IEEE,68(1), 538-539, April. · doi:10.1109/PROC.1980.11686
[2] Benedetto, J.J. and Walnut, D.F. (1994). Gabor frames for L2 and related spaces, in Benedetto, J.J. and Frazier, M.W., Eds.,Wavelets: Mathematics and Applications, CRC Press, Boca Raton, FL, 97-162. · Zbl 0887.42025
[3] B?lcskei, H. and Hlawatsch, F. (1997). Discrete Zak transforms, polyphase transforms, and applications,IEEE Trans. Signal Processing,45(4), 851-866, April. · doi:10.1109/78.564174
[4] Daubechies, I. (1992).Ten Lectures on Wavelets, SIAM. · Zbl 0776.42018
[5] Daubechies, I., Landau, H.J., and Landau, Z. (1995). Gabor time-frequency lattices and the Wexler-Raz identity,J. Fourier Anal. Appl.,1(4), 437-478. · Zbl 0888.47018 · doi:10.1007/s00041-001-4018-3
[6] Feichtinger, H.G. and Strohmer, T., Eds., (1998).Gabor Analysis and Algorithms: Theory and Applications, Birkh?user, Boston, MA. · Zbl 0890.42004
[7] Gohberg, I., Lancaster, P., and Rodman, L. (1982).Matrix Polynomials, Academic Press. · Zbl 0482.15001
[8] Heil, C.E. and Walnut, D.F. (1989). Continuous and discrete wavelet transforms,SIAM Rev.,31(4), 628-666, December. · Zbl 0683.42031 · doi:10.1137/1031129
[9] Janssen, A.J.E.M. (1981). Gabor representation of generalized functions,J. Math. Anal. Appl.,83, 377-394. · Zbl 0473.46028 · doi:10.1016/0022-247X(81)90130-X
[10] Janssen, A.J.E.M. (1988). The Zak transform: A signal transform for sampled time-continuous signals,Philips J. Res.,43(1), 23-69. · Zbl 0653.94002
[11] Janssen, A.J.E.M. (1994). Signal analytic proofs of two basic results on lattice expansions,Applied and Computational Harmonic Analysis,1, 350-354. · Zbl 0834.42019 · doi:10.1006/acha.1994.1021
[12] Janssen, A.J.E.M. (1995). Duality and biorthogonality for Weyl-Heisenberg frames,J. Fourier Anal. Appl.,1(4), 403-436. · Zbl 0887.42028 · doi:10.1007/s00041-001-4017-4
[13] Janssen, A.J.E.M. (1995). On rationally oversampled Weyl-Heisenberg frames,Signal Processing,47, 239-245. · Zbl 0875.94048 · doi:10.1016/0165-1684(95)00112-3
[14] Janssen, A.J.E.M. (1998). The duality condition for Weyl-Heisenberg frames, in Feichtinger, H.G. and Strohmer, T., Eds.,Gabor Analysis and Algorithms: Theory and Applications, Birkh?user, Boston, MA, 33-84.
[15] Landau, H.J. (1993). On the density of phase-space expansions,IEEE Trans. Inf. Theory,39, 1152-1156. · Zbl 0808.94004 · doi:10.1109/18.243434
[16] Ron, A. and Shen, Z. (1995). Frames and stable bases for shift-invariant subspaces of L2(R d ),Can. J. Math.,47(5), 1051-1094. · Zbl 0838.42016 · doi:10.4153/CJM-1995-056-1
[17] Walnut, D.F. (1992). Continuity properties of the Gabor frame operator,J. Math. Anal. Appl.,165, 479-504. · Zbl 0763.47014 · doi:10.1016/0022-247X(92)90053-G
[18] Wexler, J. and Raz, S. (1990). Discrete Gabor expansions,Signal Processing,21, 207-220. · doi:10.1016/0165-1684(90)90087-F
[19] Zibulski, M. and Y.Y. Zeevi. (1993). Oversampling in the Gabor scheme,IEEE Trans. Signal Processing,41(8), 2679-2687, August. · Zbl 0800.94088 · doi:10.1109/78.229898
[20] Zibulski, M. and Zeevi, Y.Y. (1997). Analysis of multiwindow Gabor-type schemes by frame methods,Applied and Computational Harmonic Analysis,4(2), 188-221, April. · Zbl 0885.42024 · doi:10.1006/acha.1997.0209
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