Interpolation and Sampling Hypersurfaces for the Bargmann-Fock space in higher dimensions

Abstract

We study those smooth complex hypersurfaces $W$ in $\C ^n$ having the property that all holomorphic functions of finite weighted $L^p$ norm on $W$ extend to entire functions with finite weighted $L^p$ norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces $W$ in $\C ^n$ such that any entire function with finite weighted $L^p$ norm is stably determined by its restriction to $W$. We provide sufficient geometric conditions on the hypersurface to be an interpolation and sampling hypersurface. The geometric conditions that imply the extension property and the restriction property are given in terms of some directional densities.