Sampling and interpolation in de Branges spaces with doubling phase

TL;DR

This paper characterizes sampling and interpolation in de Branges spaces with doubling phase derivatives using geometric density conditions, extending techniques from Fock spaces to these more general spaces.

Contribution

It provides a geometric density criterion for sampling and interpolation in de Branges spaces with doubling phase, linking these spaces to one component model spaces.

Findings

Established density conditions for sampling and interpolation.

Connected de Branges spaces with doubling phase to one component model spaces.

Applied techniques from Fock space analysis to de Branges spaces.

Abstract

The de Branges spaces of entire functions generalise 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 characterisation, 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 one component model spaces. A novelty of our work is the application to de Branges spaces of techniques developed by Marco, Massaneda and Ortega-Cerd\'a for Fock spaces satisfying a doubling condition analogue to ours.