2026/07/21 by Luís Daniel Abreu, Michael Speckbacher · 2 citations
#math.CA #math.FA
Let Ω⊂ ℝ be a bounded interval and let PW(Ω) be the corresponding Paley--Wiener space. For a measurable set E⊂ ℝ of finite measure, consider the largest possible fraction of the L2-mass of a function in PW(Ω) that can lie in E. We prove that this concentration is no larger than the concentration attained on an interval of measure | E| . Thus, intervals optimize concentration in the Paley-Wiener space of band-limited functions. The proof, based on an universality-type limit of the reproducing kernel of analytic trigonometric polynomials on the circle, has two steps. First, we establish an optimal concentration theorem for analytic trigonometric polynomials on the circle. Second, the universality-type limit transfers the result from the circle to the real line, by controlling the expansion of circles whose projection kernels are midpoint Riemann sums for the Paley--Wiener sinc kernel.