On the Laplace transform of distributions of the form T(P ± io, λ)

Let f(z, lambda), z is an element of C, be an entire function of the variables z, lambda,f(z, lambda) = Sigma(nu=0)(infinity) alpha(nu)(lambda)z(nu). Let us consider the family of distributions of the form [1], p. 285, [GRAPHICS] In this paper we generalize the concept of the classic one-dimensional Laplace transform to certain distributions (P +/- io)(lambda) (an important contribution of Gelfand, cf. [1], p. 274). We define the causal, anticausal distributions R-alpha (P +/- io,n) (cf. [3.1], p. 4). This distribution is an elementary solution of the homogeneous ultrahyperbolic operator iterated k -times. We observe that the distributional function R-alpha(P +/- io, n) is a (causal, anticausal) analogue of the kernel due to A. P. Calderon, Aronszjan-Smith and L. Schwartz (cf. M.I.T. [1958], Ann. Inst. Fourier [1961] and Hermann, Paris [1960], respectively).