Reproducing Kernels for Hilbert Spaces of Real Harmonic Functions

Abstract

This paper introduces a family of Hilbert spaces of real harmonic functions on bounded regions in $\mathbb{R}^n$ and shows that, for a range of values of s, they are reproducing kernel Hilbert spaces. The spaces are characterized by their boundary traces, and the inner products are defined via their expansions in the harmonic Steklov eigenfunctions of the region. The reproducing kernels are described explicitly in terms of the Steklov eigenfunctions. Expansions for some of the standard integral operators defining the solutions of Dirichlet, Robin, and Neumann boundary value problems for Laplace's equation are derived, and relationships of these operators to some reproducing kernels are described.