Linear independence of time-frequency translates

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