Linear independence of time-frequency translates
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- by Christopher Heil, Jayakumar Ramanathan and Pankaj Topiwala
- Proc. Amer. Math. Soc. 124 (1996), 2787-2795
- DOI: https://doi.org/10.1090/S0002-9939-96-03346-1
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Abstract:
The refinement equation $\varphi (t) = \sum _{k=N_1}^{N_2} c_k \varphi (2t-k)$ plays a key role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates $|a|^{1/2} \varphi (at-b)$ of $\varphi \in L^2(\mathbf {R})$, it is natural to ask if there exist similar dependencies among the time-frequency translates $e^{2 \pi i b t} f(t+a)$ of $f \in L^2(\mathbf {R})$. In other words, what is the effect of replacing the group representation of $L^2(\mathbf {R})$ induced by the affine group with the corresponding representation induced by the Heisenberg group? This paper proves that there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is proved that for each arbitrary finite collection $\{(a_k,b_k)\}_{k=1}^N$, the set of all functions $f \in L^2(\mathbf {R})$ such that $\{e^{2 \pi i b_k t} f(t+a_k)\}_{k=1}^N$ is independent is an open, dense subset of $L^2(\mathbf {R})$. It is conjectured that this set is all of $L^2(\mathbf {R}) \setminus \{0\}$.References
- J. Benedetto, C. Heil, and D. Walnut, Differentiation and the Balian–Low Theorem, J. Fourier Anal. Appl. 1 (1995), 355–402.
- Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, DOI 10.1090/memo/0453
- O. Christensen, Frames containing a Riesz basis and approximation of the frame coefficients using finite-dimensional methods, preprint (1995).
- O. Christensen and C. Heil, Perturbations of frames and atomic decompositions, Math. Nachr. (to appear).
- Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), no. 5, 961–1005. MR 1066587, DOI 10.1109/18.57199
- Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
- Hans G. Feichtinger and Karlheinz Gröchenig, A unified approach to atomic decompositions via integrable group representations, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 52–73. MR 942257, DOI 10.1007/BFb0078863
- Hans G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. II, Monatsh. Math. 108 (1989), no. 2-3, 129–148. MR 1026614, DOI 10.1007/BF01308667
- Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
- Christopher E. Heil and David F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), no. 4, 628–666. MR 1025485, DOI 10.1137/1031129
- A. J. E. M. Janssen, The Zak transform: a signal transform for sampled time-continuous signals, Philips J. Res. 43 (1988), no. 1, 23–69. MR 947891
- Gerald Kaiser, Deformations of Gabor frames, J. Math. Phys. 35 (1994), no. 3, 1372–1376. MR 1262751, DOI 10.1063/1.530594
- H. J. Landau, A sparse regular sequence of exponentials closed on large sets, Bull. Amer. Math. Soc. 70 (1964), 566–569. MR 206615, DOI 10.1090/S0002-9904-1964-11202-7
- S. Mann, R. G. Baraniuk, S. Haykin, and D. J. Jones, The chirplet transform: mathematical considerations, and some of its applications, IEEE Trans. Signal Proc. (to appear).
- J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comp. Harm. Anal. 2 (1995), 148–153.
- Marc A. Rieffel, von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981), no. 4, 403–418. MR 639575, DOI 10.1007/BF01465863
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
Bibliographic Information
- Christopher Heil
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160 and The MITRE Corporation, Bedford, Massachusetts 01730
- Email: heil@math.gatech.edu
- Jayakumar Ramanathan
- Affiliation: Department of Mathematics, Eastern Michigan University, Ypsilanti, Michigan 48197
- Email: ramanath@emunix.emich.edu
- Pankaj Topiwala
- Affiliation: The MITRE Corporation, Bedford, Massachusetts 01730
- Email: pnt@linus.mitre.org
- Received by editor(s): March 13, 1995
- Additional Notes: The first author was partially supported by National Science Foundation Grant DMS-9401340 and by the MITRE Sponsored Research Program. The second author was partially supported by National Science Foundation Grant DMS-9401859 and by a Faculty Research and Creative Project Fellowship from Eastern Michigan University. The third author was supported by the MITRE Sponsored Research Program.
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2787-2795
- MSC (1991): Primary 42C99, 46B15; Secondary 46C15
- DOI: https://doi.org/10.1090/S0002-9939-96-03346-1
- MathSciNet review: 1327018