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

TL;DR

This paper refines bounds on roots of functions equal to their Fourier transform, proves the existence of extremizers with infinitely many double roots, and introduces a new Hermite polynomial structure connecting pointwise evaluation to linear flows on the torus.

Contribution

It improves bounds for the root location problem, proves extremizer existence with infinitely many double roots, and establishes a novel Hermite polynomial structure linking evaluation to torus flows.

Findings

Refined root bounds for Fourier-invariant functions

Existence of extremizers with infinitely many double roots

New Hermite polynomial structure relating evaluation to torus flows

Abstract

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.