Stable phase retrieval in function spaces

TL;DR

This paper investigates stable phase retrieval in function spaces, constructing subspaces with this property, connecting it to $ ext{Lambda}(p)$-set theory, and establishing foundational results for broader analysis.

Contribution

It introduces the concept of stable phase retrieval in function spaces, constructs examples, and links it to $ ext{Lambda}(p)$-set theory, while improving stability bounds and characterizing certain function spaces.

Findings

Constructed various subspaces with stable phase retrieval

Connected stable phase retrieval to $ ext{Lambda}(p)$-set theory

Improved stability bounds for phase retrieval

Abstract

Let $(\Omega,\Sigma,\mu)$ be a measure space, and $1\leq p\leq \infty$. A subspace $E\subseteq L_p(\mu)$ is said to do stable phase retrieval (SPR) if there exists a constant $C\geq 1$ such that for any $f,g\in E$ we have $$ \inf_{|\lambda|=1} \|f-\lambda g\|\leq C\||f|-|g|\|. $$ In this case, if $|f|$ is known, then $f$ is uniquely determined up to an unavoidable global phase factor $\lambda$; moreover, the phase recovery map is $C$-Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics. In this article, we construct various subspaces doing stable phase retrieval, and make connections with $\Lambda(p)$-set theory. Moreover, we set the foundations for an analysis of stable phase retrieval in general function spaces. This, in particular, allows us to show that H\"older stable phase retrieval implies stable phase retrieval, improving the stability bounds in a recent article of M. Christ and the third and fourth authors. We also characterize those compact Hausdorff spaces $K$ such that $C(K)$ contains an infinite dimensional SPR subspace.