Abstract:
We study the problem of finding the norm of the point evaluation operator in the Paley--Wiener space PW^{1}(R^d), consisting of d-variable functions of spherical exponential type that are integrable on R^d. The extremal functions can be taken radial, which naturally leads us to consider a related extremal problem in a weighted Paley--Wiener space of single-variable functions. We establish that the radial extremal function must satisfy a third-order linear ODE with polynomial coefficients for every d≥1, extending Gorbachev's recent odd-dimensional result. Along the way, we prove interpolation and reciprocal formulas involving the zeros of the extremizer. This is joint work with Felipe Gonçalves and Danylo Radchenko.