Density theorems for sampling and interpolation in the Bargmann-Fock space. I. (English) Zbl 0745.46034

Seip, Kristian

J. Reine Angew. Math. 429, 91-106 (1992).

We give a complete description of sampling and interpolation in the Bargmann-Fock space, based on a density concept of Beurling. Roughly speaking, a discrete set is a set of sampling if and only if its density in every part of the plane is strictly larger than that of the von Neumann lattice, and similarly, a discrete set is a set of interpolation if and only if its density in every part of the plane is strictly smaller than that of the von Neumann lattice. The necessity of these conditions are proved in Part I of the paper, while Part II (reviewed above), written jointly with R. Wallstén, deals with the sufficiency.

Reviewer: K.Seip

MSC:

46E15	Banach spaces of continuous, differentiable or analytic functions
41A05	Interpolation in approximation theory
46J15	Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Keywords:

sampling and interpolation in the Bargmann-Fock space; density concept of Beurling; von Neumann lattice; necessity