LINEAR INDEPENDENCE OF TIME-FREQUENCY TRANSLATES

The refinement equation phi(t) = Sigma(k=N1)(N2) c(k) phi(2t - k) play s a key role in wavelet theory and in subdivision schemes in approxima tion theory Viewed as an expression of linear dependence among the tim e-scale translates \a\(1/2)phi(at - b) of phi is an element of L(2)(R) , it is natural to ask if there exist similar dependencies among the t ime-frequency translates e(2 pi ibt) f(t + a) of f is an element of L( 2)(R). I, other words, what is the effect of replacing the group repre sentation of L(2) (R) induced by the affine group with the correspondi ng representation induced by the Heisenberg group? This paper proves t hat there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is prove d that for each arbitrary finite collection {(a(k), b(k))}(N)(k=1), th e set of all functions f is an element of L(2)(R) such that {e(2 pi ib kt) f(t + a(k))}(N)(k=1) is independent is an open, dense subset of L( 2)(R). It is conjectured that this set is all of L(2)(R) \ {0}.