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

Abstract

We study a recent result of Bourgain, Clozel and Kahane, a version of which states that a sufficiently nice function $f:\mathbb{R}\to \mathbb{R}$ that coincides with its Fourier transform and vanishes at the origin has a root in the interval $(c,\infty )$, where the optimal $c$ satisfies $0.41\le c\le 0.64$. A similar result holds in higher dimensions. We improve the one-dimensional result to $0.45\le c\le 0.594$, 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.

26 pages, 4 figures