Weighted reproducing kernels in Bergman spaces
A major inspiration for this paper is the factorization theory developed by \textit{H. Hedenmalm} [J. Reine Angew. Math. 422, 45-68 (1991; Zbl 0734.30040)] for the standard Bergman space \(A^2\), and later generalized to the Bergman space \(A^2\) by \textit{P. Duren}, \textit{D. Khavinson}, \textit{H. S. Shapiro} and \textit{C. Sundberg} [Pac. J. Math. 157, No. 1, 37-56 (1993; Zbl 0739.30029)]; here \(A^p\), \(p\geq 1\), denotes the set of functions \(f\) which are analytic in the unit disk \(\Delta\) and satisfy \[ |f|_p= \Biggl( \int_\Delta |f(z)|^p dA(z)\Biggr)^{1/p}< \infty, \] where \(dA(z)\) is Lebesgue measure on \(\Delta\). A basic result in the factorization theory says that if \(\Lambda\) is a zero sequence of \(A^p\), there exists a unique (up to multiplication by a unimodular constant) function \(G_\Lambda\in A^p\) with \(|G_\Lambda|_p= 1\) such that \[ |f/ G_\Lambda |\leq |f|_p \] for any \(f\in A^p\) vanishing on \(\Lambda\), multiplicities taken into account in the usual way. A number of interesting properties of \(G_\Lambda\) is known, but it is difficult to find much explicit information about it, even in rather simple situations. Hedenmalm showed that for \(p= 2\) and \(\Lambda\) a finite sequence \(G_\Lambda\) is a linear combination of standard Bergman kernel functions, and thus in particular rational. The paper under review generalizes this result to \(p\geq 1\); it turns out that \(G_\Lambda\) is a Blaschke product times a rational function to the power \(2/p\). The proof of this result relies on the identification and study of the reproducing kernels associated with the weighted measure \[ d\mu_B(z)= |B(z)|^\alpha dA(z), \] \(B\) a finite Blaschke sequence. An interesting discovery is that the kernel function depends anti-analytically on \(\alpha\). The study of these kernel functions has some further applications, among other things, the rationality of the kernel function implies that \(B\) of \(d\mu B(z)\) is a finite Blaschke product, if \(B\) is assumed to be in the disk algebra.
scientific article; zbMATH DE number 731833