  We show that a real-valued function $f$ in the shift-invariant space
generated by a totally positive function of Gaussian type is uniquely
determined, up to a sign, by its absolute values $\{|f(\lambda)|: \lambda \in
\Lambda \}$ on any set $\Lambda \subseteq \mathbb{R}$ with lower Beurling
density $D^{-}(\Lambda)>2$.
