Point evaluation in Paley–Wiener spaces

Abstract

We study the norm of point evaluation at the origin in the Paley–Wiener space PWp for 0 < p < p < ∞, and we verify numerically that Cp>p/2 for 1 ≤ p < 2. We also estimate the asymptotic behavior of Cp as p → ∞ and as p → 0+. Our approach is based on expressing Cp as the solution of an extremal problem. Extremal functions exist for all 0 < p < ∞; they are real entire functions with only real zeros, and the extremal functions are known to be unique for 1 ≤ p < ∞. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau–Pollak–Slepian operator of time-frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range 1 < p < ∞, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to Cp and the extremal functions.