  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.
