Full Line Integro-Differential Equations of Convolution Type with Rational Symbols and Realization

摘要

The report concerns integro-differential equations of the form: (A)--q(i(d/dt)) phi (t) + integral from minus infinity to plus infinity, of k(t-s) phi (s)ds = f(t), ta member of R where q is an mxm matrix polynomial and k is an L(sub 1)(sup mxm)(R)-function with a rational Fourier transform. By definition the symbol W of equation (A) is the function W(lambda)=q(lambda)+(K circumflex)(lambda), where (K circumflex) is the Fourier transform of k. The assumptions on q and k imply that W is a rational mxm matrix function without poles on the real axis. The aim is to solve equation (A) in spaces of (possibly generalized) functions and to analyze its solutions in terms of the matrices G, A, B and C appearing in a representation of W.