A convolution product for the solutions of partial difference equations

This paper is concerned with linear partial difference operators L having constant coefficients. The functions considered are defined only on the lattice points of the complex plane. It is shown that with any two solutions of Lu(z) = 0 there is associated a new solution which is represented as a convolution product. This product may be considered as a type of line integral and is based upon a discrete analogue of Green's formula. This development may be regarded as an anology to the pioneering work of H. Lewy concerning the composition of solutions of partial differential equations. It may also be considered a continuation of the investigation pursued by Duffin and Duris in the introduction of a convolution product for discrete analytic functions. HUNT LIBRARY CARNEGIE-MELLON UNIVERSITY A Convolution Product for the Solutions of Partial Difference Equations* by R. J. Duffin and Joan Rohrer CARNEGIE INSTITUTE OF TECHNOLOGY