## Zero sequences for Bergman spaces.(English)Zbl 0871.30004

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)

zero sequence
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