ON THE RIEMANN MATRIX OF A HYPERBOLIC SYSTEM
WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER
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Consideration is given to linear first order partial differential equations with constant coefficients, which are hyperbolic. The structure and behavior of their elementary solutions are examined. The study is confined to systems which are homogeneous of the first order in derivatives, and enables one to analyze, separately, two aspects differential and matrix of the operators in the differential system. The Riemann matrix the matrix elementary solution of the system is derived from a spherical mean of the spectral resolution of the coefficient matrix. An alternative approach, based on the convergence theory of Fourier series and integrals, is used. The relation between a first order system and the irreducible equation of higher order satisfied by certain combinations of the components is examined. The Riemann matrix is then expressed as an integral over the velocity or spectral surface related to the matrix operator.