Szegő kernel

Summary

Let $\Omega \subset \C^n$ be a bounded domain with smooth boundary. Its Szegő kernel is the reproducing kernel of the Hardy space of holomorphic functions with square-integrable boundary values on $\partial\Omega$. It plays for boundary integrals the role that the Bergman kernel plays for volume integrals. It gives a canonical integral formula that recovers a holomorphic function inside $\Omega$ from its values on the boundary. On the unit disc it is the Cauchy kernel, and on the unit ball it has an explicit closed form.