Norm expansion along a zero variety in ${\mathbb C}^d$

Abstract

The reproducing kernel function of a weighted Bergman space over domains in ${\mathbb C}^d$ is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension $d=2$ for certain classes of weighted Bergman spaces over the bidisk (with the diagonal $z_1=z_2$ as subvariety) and the ball (with $z_2=0$ as subvariety), as well as for a weighted Bargmann-Fock space over ${\mathbb C}^2$ (with the diagonal $z_1=z_2$ as subvariety).