Two complete and minimal systems associated with the zeros of the Riemann zeta function

Abstract

We link together three themes which had remained separated so far: the Hilbert space properties of the Riemann zeros, the ``dual Poisson formula'' of Duffin-Weinberger (also named by us co-Poisson formula), and the ``Sonine spaces'' of entire functions defined and studied by de Branges. We determine in which (extended) Sonine spaces the zeros define a complete, or minimal, system. We obtain some general results dealing with the distribution of the zeros of the de Branges Sonine entire functions. We draw attention onto some distributions associated with the Fourier transform and which we introduced in our earlier works.