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 $C<\mathrm{\infty }$ and $\gamma \in (0,1]$ such that $$\underset{|z|=1}{min}\Vert f-zg{\Vert }_{L^2}\le C\Vert |f|-|g|{\Vert }_{L^2}^{\gamma }(\Vert f{\Vert }_{L^2}+\Vert g{\Vert }_{L^2}{)}^{1-\gamma }\mathrm{\forall }f,g\in V,(\ast )$$

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.