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Phase retrieval is known to always be unstable when using a frame orcontinuous frame for an infinite dimensional Hilbert space. We consider ageneralization of phase retrieval to the setting of subspaces of $L_2$ whichcoincides with using a continuous frame for phase retrieval when the subspaceis the range of the analysis operator of a continuous frame. We then prove thatthere do exist infinite dimensional subspaces of $L_2$ where phase retrieval isstable. That is, we give a method for constructing an infinite dimensionalsubspace $Y\subseteq L_2$ such that there exists $C\geq 1$ so that$$\min\big(\big\|f-g\big\|_{L_2},\big\|f+g\big\|_{L_2}\big)\leq C \big\||f|-|g| \big\|_{L_2} \qquad\textrm{ for all }f,g\in Y. $$  This construction also leads to new results on uniform stability of phaseretrieval in finite dimensions. Our construction has a deterministic componentand a random component. When using sub-Gaussian random variables we achievephase retrieval with high probability and stability constant independent of thedimension $n$ when using $m$ on the order of $n$ random vectors. Withoutsub-Gaussian or any other higher moment assumptions, we are able to achievephase retrieval with high probability and stability constant independent of thedimension $n$ when using $m$ on the order of $n\log(n)$ random vectors.

Stable phase retrieval for infinite dimensional subspaces of $L_2(\mathbb{R})$