Reproducing Kernels for Hilbert Spaces of Real Harmonic Functions

This paper studies a family of Hilbert spaces of real harmonic functions on bounded regions in Rn and will show 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 will then be 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 will also be derived and relationships of these operators to some reproducing kernels will be described