CONTINUOUS-SPECTRUM EIGENFUNCTION-EXPANSIONS AND THE CAUCHY-PROBLEM IN L1

We consider the Cauchy problem a partial derivative PSI = -tauPSI, PSI (x, 0) = f(x), where T is a ordinary differential operator in x (of or der at least 2), x belongs to an unbounded interval I subset-of R, and f is-an-element-of L1(I). The fact that f does not belong to L2(I) to gether with the general nature of the differential expression tau prec lude the application of classical methods in the case where the order of tau is more than 2; instead we use a continuous spectrum eigenfunct ion expansion, developed in the paper, to obtain a solution PSI(f) of the problem which depends continuously on f in an appropriate sense an d also converges to f in a natural topology as t --> 0. The solution d epends upon a kernel function which may, in particular cases, be calcu lated explicitly. Questions of approximation of the solution by finite sums are also considered.