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.
Original language | English |
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Pages (from-to) | 678-711 |
Journal | Journal of Mathematical Analysis and Applications |
Volume | 451 |
Issue number | 2 |
Early online date | 21 Feb 2017 |
DOIs | |
Publication status | Published - 15 Jul 2017 |
Bibliographical note
26 pages, 4 figuresKeywords
- math.CA
- 33C45, 42B10