A Blaschke-type product and random zero sets for Bergman spaces. (English) Zbl 0764.30029

This paper provides significant new information about zero sets of analytic functions which are in the Bergman spaces \(\mathbb{A}^ p\) of functions in the unit disc. For each \(\lambda\), \(|\lambda|<1\) a Blaschke-type factor \(f_ \lambda\) is introduced which vanishes at \(\lambda\) and is a contractive zero divisor in \(\mathbb{A}^ 2\): \(\| g/f\|_ 2\leq\| g\|_ 2\) whenever \(g\in\mathbb{A}^ 2\), \(g(\lambda)=0\). Further, if \(f\not\equiv 0\in\mathbb{A}^ 2\), then the infinite product \(B\), of the \(f_ \lambda\), as \(\lambda\) ranges over the zeros of \(f\), converges. Thus we have \(f=BF\), with \(F\neq 0\), \(\| F\|_ 1\leq\| f\|_ 2\).
The main result, Theorem 2.11, asserts that if a discrete sequence \(\Lambda=\{\lambda_ n\}\) grows sufficiently slowly, then for almost all independent choices of \(\{\theta_ n\}\), the sequence \(\{\lambda_ n e^{i\theta_ n}\}\) is an \(\mathbb{A}^ p\)-zero set, if \(1\leq p\leq 2\) (for \(p>2\) slightly less precise conditions are given). The author poses interesting conjectures in §3 concerning the best possible growth allowed of such \(\Lambda\), and a refinement of a zero-one law. The paper contrasts this work with earlier work of C.Horowitz [Duke Math. J. 41, 693-710 (1974; Zbl 0293.30035)] and H. Hedenmalm [J. Reine Angew. Math. 422, 45-68 (1991; Zbl 0734.30040)]. The Blaschke-type product introduced here differ from those of Hedenmalm that the latter considers an extremal problem over a full sequence \(\Lambda\), rather than as a product of individual factors.


30D50 Blaschke products, etc. (MSC2000)
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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