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Linear independence of time-frequency translates

By Christopher Heil, Jayakumar Ramanathan and Pankaj Topiwala


Abstract. The refinement equation ϕ(t) = �N2 k=N1 ckϕ(2t − k) playsakey role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates |a | 1/2ϕ(at − b)ofϕ∈L2 (R), it is natural to ask if there exist similar dependencies among the time-frequency translates e2πibtf(t + a) off∈L2 (R). In other words, what is the effect of replacing the group representation of L2 (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 {(ak,bk)} N k=1, the set of all functions f ∈ L2 (R) such that {e2πibktf(t+ ak)} N k=1 is independent is an open, dense subset of L2 (R). It is conjectured that this set is all of L2 (R) \{0}. 1

Year: 1996
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