Continuous Spectrum Eigenfunction Expansions and the Cauchy Problem in L1

We consider the Cauchy problem partial Psi/partial t = -τ Psi, Psi(x, 0) = f(x), where τ is a ordinary differential operator in x (of order at least 2), x belongs to an unbounded interval [Note: See the image of page 407 for this formatted text] I subset R, and fin L₁(I). The fact that f does not belong to L₂(I) together with the general nature of the differential expression τ preclude the application of classical methods in the case where the order of τ is more than 2; instead we use a continuous spectrum eigenfunction expansion, developed in the paper, to obtain a solution Psi_f of the problem which depends continuously on f in an appropriate sense and also converges to f in a natural topology as t -> 0. The solution depends upon a kernel function which may, in particular cases, be calculated explicitly. Questions of approximation of the solution by finite sums are also considered.