Sign Retrieval in Shift-Invariant Spaces with Totally Positive Generator

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 $$\{|f(\lambda )|: \lambda \in \Lambda \}$$ { | f ( λ ) | : λ ∈ Λ } on any set $$\Lambda \subseteq {\mathbb {R}}$$ Λ ⊆ R with lower Beurling density $$D^{-}(\Lambda )>2$$ D - ( Λ ) > 2 .We consider a totally positive function of Gaussian type, i.e., a function $$g \in L^2({\mathbb {R}})$$ g ∈ L 2 ( R ) whose Fourier transform factors as $$\begin{aligned} \hat{g}(\xi )= \int _{{\mathbb {R}}} g(x) e^{-2\pi i x \xi } dx = C_0 e^{- \gamma \xi ^2}\prod _{\nu =1}^m (1+2\pi i\delta _\nu \xi )^{-1}, \quad \xi \in {\mathbb {R}}, \end{aligned}$$ g ^ ( ξ ) = ∫ R g ( x ) e - 2 π i x ξ d x = C 0 e - γ ξ 2 ∏ ν = 1 m ( 1 + 2 π i δ ν ξ ) - 1 , ξ ∈ R , with $$\delta _1,\ldots ,\delta _m\in {\mathbb {R}}, C_0, \gamma >0, m \in {\mathbb {N}} \cup \{0\}$$ δ 1 , … , δ m ∈ R , C 0 , γ > 0 , m ∈ N ∪ { 0 } , and the shift-invariant space$$\begin{aligned} V^\infty (g) = \Big \{ f=\sum _{k \in {\mathbb {Z}}} c_k\, g(\cdot -k): c \in \ell ^\infty ({\mathbb {Z}}) \Big \}, \end{aligned}$$ V ∞ ( g ) = { f = ∑ k ∈ Z c k g ( · - k ) : c ∈ ℓ ∞ ( Z ) } , generated by its integer shifts within $$L^\infty ({\mathbb {R}})$$ L ∞ ( R ) . As a consequence of (1), each $$f \in V^\infty (g)$$ f ∈ V ∞ ( g ) is continuous, the defining series converges unconditionally in the weak$$^*$$ ∗ topology of $$L^\infty $$ L ∞ , and the coefficients $$c_k$$ c k are unique [6, Theorem 3.5].