The S.V.D. of the Poisson Kernel

TL;DR

This paper analyzes the singular value decomposition of the Poisson kernel for the Dirichlet problem, relating it to eigenproblems, characterizing harmonic projections, and providing optimal approximations and spectral formulas.

Contribution

It introduces the SVD of the Poisson kernel, connects it to eigenproblems, and derives bounds and approximations for harmonic analysis in bounded regions.

Findings

Singular values and functions related to Dirichlet Biharmonic Steklov eigenproblem.

Characterization of the Bergman harmonic projection and reproducing kernel.

Optimal finite rank approximations with error estimates and spectral formulas.

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.