Stability of concentration in the Paley-Wiener space

Abstract

We prove stability of concentration in the Paley-Wiener space for each fixed time-bandwidth product. The concentration deficit controls the squared $L^2(\mathbb R)$-distance from a translated first prolate spheroidal wave function and the squared normalized symmetric difference from an interval at the same center. The obtained exponent for the function distance is optimal. The main idea is the introduction of the set discrepancy weighted by the best optimizers in place of the symmetric difference. The proof then combines elementary Hilbert space arguments with classical properties of the prolate spheroidal wave functions.