Beurling type density theorems in the unit disk
We consider two equivalent density concepts for the unit disk that provide a complete description of sampling and interpolation in \(A^{- n}\) (the Banach space of functions \(f\) analytic in the unit disk with \((1-| z |^ 2)^ n | f(z) |\) bounded). This study reveals a `Nyquist density': A sequence of points is (roughly speaking) a set of sampling if and only if its density in every part of the disk is strictly larger than \(n\), and it is a set of interpolation if and only if its density in every part of the disk is strictly smaller than \(n\). Similar density theorems are also obtained for weighted Bergman spaces.
scientific article; zbMATH DE number 549679
scientific article