Difference equations in several variables

In this paper we shall derive energy integral inequalities for the linear hyperbolic partial differential equation of any order in several variables with multiple characteristics and constant coefficients. In [1] Garding uses an idea by Leray [6] to develop energy integrals for totally hyperbolic equations (including variable coefficients). Totally hyperbolic equations are subject to the severe restriction that all characteristics remain distinct(2). In his method G'arding multiplies the given operator of order n by another operator of order n 1 whose characteristics separate those of the given operator. Using the Fourier transform and Parseval's formula he arrives at the energy inequality, which is an a priori estimate of the square integral of the solution of the given equation and all the derivatives up to order n -1. In our case of an equation with multiple characteristics we use a variant of G'arding's method. Our energy inequality is of a different form in as much as it provides an a priori estimate of the integral of the square of the solution and certain non-negative integral forms of its derivatives up to order n -1. However, we also show that for totally hyperbolic equations our energy inequalities reduce to the standard form. In hyperbolic equations with multiple characteristics the influence of the lower order terms must be taken into account and hence the form of these lower order terms is suitably restricted. In totally hyperbolic equations no restrictions are imposed on the lower order terms. Our definition of hyperbolicity is therefore such that in the case of distinct characteristics no restrictions are imposed on the lower order terms. In [2] Gairding has given a criterion for hyperbolicity of equations in several variables with multiple characteristics and constant coefficients. A. Lax [4] extended this criterion, in the case of two independent variables, to include variable coefficients. We shall not be concerned here with

Spectral synthesis and spectral analysis.- Spectral analysis and spectral synthesis on discrete Abelian groups.- Spectral synthesis and functional equations.- Mean periodic functions.- Difference equations in several variables.- Spectral analysis and synthesis on polynomial hypergroups in a single variable.- Spectral analysis and synthesis on multivariate polynomial hypergroups.

Discrete orthogonal polynomials and difference equations of several variables