{"id":504,"date":"2022-05-08T19:47:24","date_gmt":"2022-05-09T02:47:24","guid":{"rendered":"\/wp\/hades\/?p=504"},"modified":"2022-05-08T19:47:24","modified_gmt":"2022-05-09T02:47:24","slug":"examples-of-holder-stable-phase-retrieval","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/hades\/2022\/05\/08\/examples-of-holder-stable-phase-retrieval\/","title":{"rendered":"Examples of H\u00f6lder-Stable Phase Retrieval"},"content":{"rendered":"\n\t\t\t\t\n<p>The HADES seminar on Tuesday, <strong>May 10<\/strong>\u00a0will be at\u00a0<strong>3:30 pm<\/strong>\u00a0in\u00a0<strong>Room 1015 <\/strong>(Notice the room change).<\/p>\n\n\n\n<p><strong>Speaker:<\/strong> Benjamin Pineau<\/p>\n\n\n\n<p><strong>Abstract:<\/strong> 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 $C &lt; \\infty$ and $\\gamma \\in (0, 1]$ such that \\begin{equation}\\label{eq} \\min_{|z|=1} \\|f \u2212 zg\\|_{L^2} \\leq C\\||f| \u2212 |g|\\|_{L^2}^\\gamma (\\|f\\|_{L^2} + \\|g\\|_{L^2} )^{1\u2212\u03b3}\\,\\forall f, g \\in V,(*)\\end{equation}<\/p>\n\n\n\n<p>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 $\\hat F$.<\/p>\n\n\n\n<p>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.<\/p>\n\t\t","protected":false},"excerpt":{"rendered":"<p>The HADES seminar on Tuesday, May 10\u00a0will be at\u00a03:30 pm\u00a0in\u00a0Room 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 [&hellip;]<\/p>\n","protected":false},"author":90,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-504","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/504","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/users\/90"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/comments?post=504"}],"version-history":[{"count":0,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/504\/revisions"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/media?parent=504"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/categories?post=504"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/tags?post=504"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}