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

Abstract

Abstract. 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.

An optimal uncertainty principle in twelve dimensions via modular forms

“…Existence of extremizers. The existence proof for extremizers with s = −1 is almost identical to the proof of the +1 case in [12,Section 6]. We briefly outline the proof here for completeness.…”

“…For completeness, we state our next theorem for both ±1 cases, although all the results in the following theorem were already proved for the +1 case by Gonçalves, Oliveira e Silva, and Steinerberger in [12]. Note that we regard A +1 and A −1 as synonymous with A + and A − , respectively.…”

“…Choose λ > 0 and let g(x) = f (λx). Then g(ξ) = λ −12 f (ξ/λ), and it follows that g ∈ A + (12). Moreover, if λ is close enough to 1, then r(g) and r( g) are both less than √ 2.…”

“…This section is devoted to the proof of Theorem 1.4. We deal only with the −1 case, because all the assertions in this theorem were already proved in [12] for the +1 case. First, we reduce determining A − (d) to solving Problem 1.3. such that g = −g, g(0) = 0, and r(g) ≤ r(f )r( f ).…”

“…Thus, the uncertainty principle amounts to saying that r(f ) and r( f ) cannot both be made arbitrarily small if f ∈ A + (d) \ {0}. Gonçalves, Oliveira e Silva, and Steinerberger [12,Theorem 3] proved that for each dimension d there exists a radial function f ∈ A + (d) \ {0} such that f = f and r(f ) = r( f ) = A + (d); furthermore, A + (d) is exactly the minimal value of r(g) in the following optimization problem: Problem 1.1 (+1 eigenfunction uncertainty principle). Minimize r(g) over all g : R d → R such that (1) g ∈ L 1 (R d ) \ {0} and g = g, and (2) g(0) = 0 and g is eventually nonnegative.…”