Sign retrieval in shift-invariant spaces with totally positive generator

TL;DR

This paper demonstrates that functions in certain shift-invariant spaces generated by totally positive functions of Gaussian type can be uniquely identified (up to sign) solely from their absolute values on sufficiently dense sets.

Contribution

It establishes a unique phase retrieval result for functions in shift-invariant spaces generated by totally positive functions of Gaussian type.

Findings

Functions are uniquely determined by absolute values on dense sets

The result applies to sets with lower Beurling density greater than 2

Provides a phase retrieval framework for specific shift-invariant spaces

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