We study a recent result of Bourgain, Clozel and Kahane, a version of which
states that a sufficiently nice function f:R→R
that coincides with its Fourier transform and vanishes at the origin has a root
in the interval (c,∞), where the optimal c satisfies 0.41≤c≤0.64. A similar result holds in higher dimensions. We improve the
one-dimensional result to 0.45≤c≤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.Comment: 26 pages, 4 figure