Hermite polynomials, linear flows on the torus, and an uncertainty principle for roots

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Colleges, School and Institutes


We study a recent result of Bourgain, Clozel and Kahane, a version of which states that a sufficiently nice function $f:\mathbb{R} \rightarrow \mathbb{R}$ that coincides with its Fourier transform and vanishes at the origin has a root in the interval $(c, \infty)$, where the optimal $c$ satisfies $0.41 \leq c \leq 0.64$. A similar result holds in higher dimensions. We improve the one-dimensional result to $0.45 \leq c \leq 0.594$, and the lower bound in higher dimensions. We also prove that extremizers exist, and have infinitely many double roots. With this purpose in mind, we establish a new structure statement about Hermite polynomials which relates their pointwise evaluation to linear flows on the torus, and applies to other families of orthogonal polynomials as well.

Bibliographic note

26 pages, 4 figures


Original languageEnglish
Pages (from-to)678-711
JournalJournal of Mathematical Analysis and Applications
Issue number2
Early online date21 Feb 2017
Publication statusPublished - 15 Jul 2017


  • math.CA, 33C45, 42B10