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