Zero sequences for Bergman spaces. (English) Zbl 0871.30004

Luecking, Daniel H.

Complex Variables, Theory Appl. 30, No. 4, 345-362 (1996).

Summary: Let \({\mathcal Z}\) be a sequence of points in the unit disk. For a variety of analytic function spaces, it is shown that the properties of the function \[ k_{\mathcal Z}(z)= {|z|^2 \over 2} \sum_{a\in {\mathcal Z}} {(1-|a|^2)^2 \over|1- \overline az |^2} \] completely determine whether \({\mathcal Z}\) is a zero sequence for the given space. For example, if \(A^{-n}= \{f\) analytic in \(D:f(z) (1-|z |^2)^n\) is bounded}, then \({\mathcal Z}\) is a zero sequence for \(A^{-n}\) if and only if \(k_{\mathcal Z} (z)-n \log[1/(1- |z|^2)]\) has a harmonic majorant.

MSC:

30C15

Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)

Keywords:

zero sequence