Stability of phase retrievable bm$p$-frames

Abstract

The phase retrieval problem presents itself in many applications in physics and engineering. Let $X$ be a Banach space and $1<p<\\infty$. Let $\\Phi=\\{x_n\\}_{n\\in~I}$ be a $p$-frame for $X$. We say $\\Phi$ is phase retrievable, if the equalities $|x^*(x_n)|=|y^*(x_n)|$ for every $n\\in~I$ imply that there exists $|\\alpha|=1$ so that $x^*=\\alpha~y^*$ for any $x^*,~y^*\\in~X^*$. In this paper, we prove that phase retrieval is always stable for $p$-frames in finite-dimensional Banach spaces and it is never uniformly stable for $p$-frames in an infinite-dimensional Banach space $X$ with a Schauder basis. We establish that stable phase retrieval is possible for the elements of $X^*$ that can be approximated sufficiently well by finite-dimensional expansion.