Sampling and interpolation in de Branges spaces with doubling phase

Abstract

The de Branges spaces of entire functions generalize the classical Paley-Wiener space of square summable bandlimited functions. Specifically, the square norm is computed on the real line with respect to weights given by the values of certain entire functions. For the Paley-Wiener space, this can be chosen to be an exponential function where the phase increases linearly. As our main result, we establish a natural geometric characterization in terms of densities for real sampling and interpolating sequences in the case when the derivative of the phase function merely gives a doubling measure on the real line. Moreover, a consequence of this doubling condition is that the spaces we consider are model spaces generated by a one-component inner function. A novelty of our work is the application to de Branges spaces of techniques developed by Marco, Massaneda and Ortega-Cerdà for Fock spaces satisfying a doubling condition analogous to ours.