On extending the domain of convergence of discrete harmonic kernels

Abstract

In 1963 the study of reproducing discrete harmonic kernels in a finite dimensional Hilbert space was introduced by Deeter and Springer. In the paper by Deeter and Springer many of the theorems concerning the ordinary harmonic kernel were shown to have discrete analogues. They also proved, using Fourier series methods, the convergence of the discrete harmonic kernel on a rectangular domain. The major purpose of this paper is to extend the convergence of the discrete harmonic kernel to more general domains using some general methods developed by Bramble and Hubbard. We show that the discrete harmonic kernel converges on domains which are composed of finite unions of rectangles. A proof of the convergence of the discrete Neumann problem using the representation of the Neumann function as a sum of the discrete Green's function and the discrete harmonic kernel is presented for certain restricted regions. Also given is a method of extending the convergence to regions bounded by a smooth curve provided that certain conditions on the kernel are satisfied.