Sign retrieval in shift-invariant spaces with totally positive generator

Abstract: 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 $D^{-}(\Lambda)>2$.