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Some conditions on Bergman space zero sets. (English) Zbl 0795.30006

The author extends work of Korenblum and Hedenmalm on the zero sets of the functions in the Bergman spaces \(A^ p\), with concentration on the individual spaces \(A^ p\) rather than on their union. The space \(A^ p\), \(0< p<\infty\) is the collection of functions \(f\) analytic in the unit disk \(U\) and satisfying \[ \| f\|^ p_ p= \textstyle{{1\over \pi}}\int_ U \int | f(z)|^ p dx dy< \infty. \] The author narrows the gap between conditions known to be necessary for a sequence to be the zero set of a function in \(A^ p\) and conditions known to be sufficient. The principle sufficient conditions restrict the number of sequence elements in (roughly) truncated Stolz angles at the boundary of the unit disk. The functions constructed are infinite canonical products with carefully chosen convergence factors.
Reviewer: J.W.Cannon (Provo)

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D50 Blaschke products, etc. (MSC2000)
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References:

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