[35] (with S. Bergman) Kernel functions and conformal mapping

In this well-known and widely cited paper, the authors return to their study of the Bergman reproducing kernel \(K(z,\overline{\zeta })\) in the Hilbert space of square integrable analytic functions on a bounded plane domain B whose boundary C consists of finitely many real analytic closed curves. The kernel \(K(z,\overline{\zeta })\) becomes strongly unbounded whenever z and ζ tend to the same boundary point in C, so the standard theory of integral equations does not apply. To circumvent this difficulty, the authors consider the regularized (Hermitian) kernel \(K(z,\overline{\zeta }) - \Gamma (z,\overline{\zeta }),\) where which is shown to be regular in the closed region B ∪ C and thus amenable to study via the classical theory.