Frame Phase-Retrievability and Exact Phase-Retrievable Frames

Abstract

A phase-retrievable frame $$\{f_{i}\}_{i}^{N}$$ for an n-dimensional Hilbert space is exact if it fails to be phase-retrievable when removing any element from the frame sequence. Unlike exact frames, exact phase-retrievable frames could have different lengths. We shall prove that for the real Hilbert space case, exact phase-retrievable frame of length N exists for every $$2n-1\le N\le n(n+1)/2$$. For arbitrary frames we introduce the concept of redundancy with respect to its phase-retrievability and the concept of frames with exact PR-redundancy. We investigate the phase-retrievability by studying its maximal phase-retrievable subspaces with respect to a given frame which is not necessarily phase-retrievable. These maximal PR-subspaces could have different dimensions. We are able to identify the one with the largest dimension, which can be considered as a generalization of the characterization for phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace, then $$|supp(x)| \ge k$$ for every nonzero vector $$x\in M$$. Moreover, if $$1\le k< [(n+1)/2]$$, then a k-dimensional PR-subspace is maximal if and only if there exists a vector $$x\in M$$ such that $$|supp(x) | = k$$.