The S.V.D. of the Poisson Kernel

Abstract

This paper describes the singular value decomposition (SVD) of the Poisson kernel for the Dirichlet problem for the Laplacian on bounded regions in R^N, N >=2. This operator is a compact linear transformation from L^2 of the boundary to L^2 of the region. These singular values and functions are related to the eigenvalues and eigenfunctions of the Dirichlet Biharmonic Steklov eigenproblem. The Bergman harmonic projection on L^2 is characterized and the Reproducing kernel for the real harmonic Bergman space is described. Optimal finite rank approximations of the Poisson kernel, with error estimates, are found. Spectral formulae for the normal derivatives of eigenfunctions of the Dirichlet Laplacian are found and yield bounds on a constant in an inequality of Hassell and Tao.