A probabilistic zero set condition for the Bergman space
The Bergman space is defined as the set of holomorphic functions in the unit disk for which the square of the modulus is area integrable. It is known that the Blaschke condition \((\sum^{\infty}_{j=1}(1-r_ j)<\infty)\) is the weakest condition on a sequence of numbers \(\{r_ j\}^{\infty}_{j=1}\subset (0,1)\) that ensures that any sequence in the disk with moduli \(r_ j\) is the zero sequence for some Bergman space function in the unit disk. However, there exist Bergman space functions whose zeros do not satisfy the Blaschke condition. This paper presents a condition weaker than the Blaschke condition, namely, \[ \limsup_{\epsilon \to 0}(\sum^{\infty}_{j=1}(1-r_ j)^{1+\epsilon})/\log (1/\epsilon)<1/4 \] that guarantees that a sequence of points with moduli \(r_ j\) is almost surely a Bergman space zero set, in the following sense. If the condition is satisfied then \(\{r_ je^{i\theta_ j}\}^{\infty}_{j=1}\) is a Bergman space zero set for almost all independent choices of the \(\theta_ j\). An explicit construction of a random function with the prescribed zeros that almost surely belongs to the Bergman space is provided (using Horowitz's generalization of the Blaschke factors).
scientific article; zbMATH DE number 4181695