Treffer: 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.