Sign retrieval in spaces of variable bandwidth
Philippe Jaming, Rolando Perez
TL;DR
The paper investigates sign retrieval for spaces of variable bandwidth $PW_\Omega(A_p)$, extending classical sign retrieval from Paley–Wiener spaces to a family governed by a Sturm–Liouville operator with piecewise-constant bandwidth $p$. It proves that, for $p$ a step function with finitely many jumps, real functions $f,g\in PW_\Omega(A_p)$ with $|f|=|g|$ must satisfy $f=g$ or $f=-g$, providing two distinct proofs: a toy-model approach via reproducing kernels and a general argument using the spectral transform. The work clarifies how the RKHS structure and interval-wise decomposition interact to enforce sign retrieval, and it outlines implications for sampling and future generalizations, including open questions on discrete sampling sets and extensions beyond step-function variations. Overall, it contributes new theoretical insight into phase retrieval for variable-bandwidth spaces and identifies directions for further research, such as sampling theories and alternative space constructions (e.g., Wilson bases) that may facilitate practical implementations.
Abstract
The aim of this paper is to get a deeper understanding of the spaces of variable bandwidth introduced by Gr{ö}chenig and Klotz (What is variable bandwidth? Comm. Pure Appl. Math., 70 (2017), 2039-2083). In particular, we show that when the variation of the bandwidth is modeled by a step function with a finite number of jumps, then, the sign retrieval principle applies.