Interpolation and sampling sequences for entire functions

Abstract

We characterise interpolating and sampling sequences for the spaces of entire functions $f$ such that $f e^{-\phi}\in L^p(\C)$, $p\geq 1$ where $\phi$ is a subharmonic weight whose Laplacian is a doubling measure. The results are expressed in terms of some densities adapted to the metric induced by $\Delta\phi$. They generalise previous results by Seip for the case $\phi(z)=|z|^2$, Berndtsson and Ortega-Cerdà and Ortega-Cerdà and Seip for the case when $\Delta\phi$ is bounded above and below, and Lyubarski\u{\i} \& Seip for 1-homogeneous weights of the form $\phi(z)=|z|h(\arg z)$, where $h$ is a trigonometrically strictly convex function.