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Linear independence
of time-frequency translates


Authors: Christopher Heil, Jayakumar Ramanathan and Pankaj Topiwala
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
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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 [Enhancements On Off] (What's this?)

  • [BHW] J. Benedetto, C. Heil, and D. Walnut, Differentiation and the Balian--Low Theorem, J. Fourier Anal. Appl. 1 (1995), 355--402.
  • [CDM] A. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary Subdivision, Mem. Amer. Math. Soc. 93 (1991), 1--186. MR 92h:65017
  • [C] O. Christensen, Frames containing a Riesz basis and approximation of the frame coefficients using finite-dimensional methods, preprint (1995).
  • [CH] O. Christensen and C. Heil, Perturbations of frames and atomic decompositions, Math. Nachr. (to appear).
  • [D1] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 39 (1990), 961--1005. MR 91e:42038
  • [D2] ------, Ten Lectures on Wavelets, SIAM Press, Philadelphia, 1992. MR 93e:42045
  • [FG1] H. G. Feichtinger and K. Gröchenig, A unified approach to atomic decompositions through integrable group representations, Function Spaces and Applications, Lecture Notes in Math. (M. Cwikel et al., eds.), vol. 1302, Springer--Verlag, New York, 1988, pp. 52--73. MR 89h:46035
  • [FG2] ------, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), 307--340; Banach spaces related to integrable group representations and their atomic decompositions, II, Monatshefte für Mathematik 108 (1989), 129--148. MR 91g:43012
  • [F] G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, Princeton, NJ, 1989. MR 92k:22017
  • [HW] C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 31 (1989), 628--666. MR 91c:42032
  • [J] A. J. E. M. Janssen, The Zak transform: a signal transform for sampled time-continuous signals, Philips J. Res. 43 (1988), 23--69. MR 89g:94005
  • [K] G. Kaiser, Deformations of Gabor frames, J. Math. Phys. 35 (1994), 1372--1376. MR 95a:42047
  • [L] H. J. Landau, A sparse regular sequence of exponentials closed on large sets, Bull. Amer. Math. Soc. 70 (1964), 566--569. MR 34:6433
  • [MBHJ] 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).
  • [RS] J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comp. Harm. Anal. 2 (1995), 148--153.
  • [R] M. Rieffel, Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann. 257 (1981), 403--418. MR 84f:22010
  • [W] P. Walters, An Introduction to Ergodic Theory, Springer--Verlag, New York, 1982. MR 84e:28017

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Additional 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

DOI: https://doi.org/10.1090/S0002-9939-96-03346-1
Keywords: Affine group, frames, Gabor analysis, Heisenberg group, linear independence, phase space, refinement equations, Schroedinger representation, time-frequency, wavelet analysis
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
Article copyright: © Copyright 1996 American Mathematical Society

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