Stable phase retrieval in function spaces, Part I

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.

We will discuss how problems in phase retrieval are naturally related to classical notions in the theory of Banach lattices. Through making this connection, we may apply established methods from the subject to attack problems in phase retrieval, and conversely we hope that the ideas and questions in phase retrieval will inspire a new avenue of research in the theory of Banach lattices.

This talk is based on joint work with Benjamin Pineau, Timur Oikhberg, and Mitchell Taylor.

functional analysis

Audience: researchers in the topic

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Banach spaces webinars

Series comments: Description: Research seminar on Banach spaces and related topics

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