Reproducing kernels for Hardy spaces on multiply connected domains

Research paper by Joseph A. Ball, Kevin F. Clancey

Indexed on: 01 Mar '96Published on: 01 Mar '96Published in: Integral Equations and Operator Theory

Abstract

Associated with a boundedg-holed (g≥0) planar domainD are two types of reproducing kernel Hilbert spaces of meromorphic functions onD. We give explicit formulas for the reproducing kernel functions of these spaces. The formulas are in terms of theta functions defined on the Jacobian variety of the Schottky double of the regionD. As applications we settle a conjecture of Abrahamse concerning Nevalinna-Pick interpolation on an annulus and obtain explicit formulas for the curvature (in the sense of Cowen and Douglas) of rank 1 bundle shift operators.