Examples of Hölder-Stable Phase Retrieval

The HADES seminar on Tuesday, May 10 will be at 3:30 pm in Room 1015 (Notice the room change).

Speaker: Benjamin Pineau

Abstract: Let $(X, \mathcal A, \mu)$ be a measure space. Let $V$ be a closed subspace of the (real or complex) Hilbert space $L^2 = L^2 (\mu)$. We say that $V$ does Holder-stable phase retrieval if there exists a constant $C < \infty$ and $\gamma \in (0, 1]$ such that \begin{equation}\label{eq} \min_{|z|=1} \|f − zg\|_{L^2} \leq C\||f| − |g|\|_{L^2}^\gamma (\|f\|_{L^2} + \|g\|_{L^2} )^{1−γ}\,\forall f, g \in V,(*)\end{equation}

Recently, Calderbank, Daubechies, Freeman, and Freeman have studied real subspaces of real-valued $L^2$ for which (*) holds with $\gamma = 1$ and constructed the first examples of such infinite-dimensional subspaces. In this situation, if $|f|$ is known then $f$ is uniquely determined almost everywhere up to an unavoidably arbitrary global phase factor of $\pm 1$. Moreover, if $|f|$ is known within a small tolerance in norm then up to such a global phase factor, f is determined within a correspondingly small tolerance. This issue arises for instance in crystallography, where one seeks to recover an unknown function $F \in L^2 (\mathbb R)$ from the absolute value of its Fourier transform $\hat F$.

In this talk, I will discuss a set of simple sufficient conditions for constructing infinite-dimensional (real and complex) subspaces $V \subset L^2 (\mu)$ which satisfy (*) and show how to construct some natural examples in which (*) holds. These examples include certain variants of Rademacher series and lacunary Fourier series. This is a joint work with Michael Christ and Mitchell Taylor.

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