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