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 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 of , it is natural to ask if there exist similar dependencies among the time-frequency translates of . In other words, what is the effect of replacing the group representation of 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 , the set of all functions such that is independent is an open, dense subset of . It is conjectured that this set is all of .

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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