Über Grundlösungen von Differenzenoperatoren

Abstract

The method to solve problems of the mathematical physics using the so-called fundamental solutions and its properties is well-known. There is the question how does it work in the case of finite difference equations approximating the mentioned problems. An answer can be given treating difference equations as functional equations in certain spaces of distributions. By a suitable construction it is shown that every linear forward difference operator with constant coefficients possesses a tempered fundamental solution in a neighbourhood (with respect to the discretisation parameter) of the fundamental solution of the corresponding differential operator. An initial-value problem for the one-dimensional equation of heat conduction as an example for a useful application of discrete fundamental solutions closes the paper.