The S.V.D. of the Poisson Kernel

Abstract

The solution of the Dirichlet problem for the Laplacian on bounded regions $\Omega$ in ${\mathbb R}^{N}, \ N \ge 2$ is generally described via a boundary integral operator with the Poisson kernel. Under mild regularity conditions on the boundary, this operator is a compact linear transformation of $L^2 (\partial \Omega ,d \sigma )$ to $L^2_H(\Omega )$—the Bergman space of $L^2$-harmonic functions on $\Omega$. This paper describes the singular value decomposition of this operator and related results. The singular functions and singular values are constructed using Steklov eigenvalues and eigenfunctions of the biharmonic operator on $\Omega$. These allow a spectral representation of the Bergman harmonic projection and the identification of an orthonormal basis of the real harmonic Bergman space $L^2_H(\Omega )$. A reproducing kernel for $L^2_H(\Omega )$ and an orthonormal basis of the space $L^2 (\partial \Omega ,d \sigma )$ also are found. This enables the description of optimal finite rank approximations of the Poisson kernel with error estimates. Explicit spectral formulae for the normal derivatives of eigenfunctions for the Dirichlet Laplacian on $\partial \Omega$ are obtained and used to identify a constant in an inequality of Hassell and Tao.