A Blaschke-type product and random zero sets for Bergman spaces
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 \S3 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 \textit{C.Horowitz} [Duke Math. J. 41, 693-710 (1974; Zbl 0293.30035)] and \textit{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.
scientific article; zbMATH DE number 91590
scientific article