  We establish the linear independence of time-frequency translates for
functions $f$ having one sided decay $\lim_{x\to \infty} |f(x)| e^{cx \log x} =
0$ for all $c>0$. We also prove such results for functions with faster than
exponential decay, i.e., $\lim_{x\to \infty} |f(x)| e^{cx} = 0$ for all $c>0$,
under some additional restrictions.
