Linear independence of timefrequency translates
Authors:
Christopher Heil, Jayakumar Ramanathan and Pankaj Topiwala
Journal:
Proc. Amer. Math. Soc. 124 (1996), 27872795
MSC (1991):
Primary 42C99, 46B15; Secondary 46C15
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 timescale translates of , it is natural to ask if there exist similar dependencies among the timefrequency 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 latticetype 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 303320160 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/S0002993996033461
PII:
S 00029939(96)033461
Keywords:
Affine group,
frames,
Gabor analysis,
Heisenberg group,
linear independence,
phase space,
refinement equations,
Schroedinger representation,
timefrequency,
wavelet analysis
Received by editor(s):
March 13, 1995
Additional Notes:
The first author was partially supported by National Science Foundation Grant DMS9401340 and by the MITRE Sponsored Research Program. The second author was partially supported by National Science Foundation Grant DMS9401859 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
