Weighted Bergman kernel functions associated to meromorphic functions (Q521294)

Let \(\Omega\) be a domain in \(\mathbb C^n\). The weighted Bergman space \(A_{\varphi}^2(\Omega)\) is the space of holomorphic functions on \(\Omega\) square integrable with respect to \(\varphi(w)dV_w\), where \(dV_w\) is the real \(2n\)-dimensional Lebesgue volume measure. The author studies the reproducing kernels \(K_{\varphi}^{\Omega}(z,w)\) of the weighted Bergman spaces \(A_{\varphi}^2(\Omega)\) (mostly in the case \(n=1\)). For example, for \(\Omega\subset\mathbb C\) and a weight \(\varphi\) bounded in a neighborhood of \(c\in\Omega\), the author provides a formula for \(K_{|z-c|^2\varphi}^{\Omega}(z,w)\) in terms of \(K_{\varphi}^{\Omega}(z,w)\). As an application the author describes the relations between the zeros of the considered reproducing kernels.