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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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 $L^2(\mathbf {R}) \setminus \{0\}$.


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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: http://dx.doi.org/10.1090/S0002-9939-96-03346-1
PII: S 0002-9939(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