"Stable phase retrieval for finite and infinite dimensional subspaces"Freeman, DanielA frame (x_j) for a Hilbert space H allows for the stable reconstruction of any vector $x\in H$ from the frame coefficients. The goal of phase retrieval is to reconstruct x (up to a unimodular scalar) using only the absolute value of the frame coefficients. Phase retrieval using a frame for a finite dimensional Hilbert space is known to always be stable where as phase retrieval using a frame or a continuous frame for an infinite dimensional Hilbert space is always unstable. In this talk we introduce a generalization of phase retrieval to the setting of subspaces of Banach lattices and characterize when an infinite dimensional subspace of a real Banach lattice allows for stable phase retrieval. By further restricting to finite dimensional subspaces, these constructions provide new examples of continuous Parseval frames which do stable phase retrieval with stability constant independent of the dimension. The continuous frames can then be randomly sampled to give new constructions of random frames which with high probability do stable phase retrieval with stability constant independent of the dimension. This talk will cover joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman, joint work with Dorsa Ghoreishi, and joint work with Timur Oikhberg, Ben Pineau, and Mitchell Taylor. |
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