On the Fourier Transform of Causal Distributions

We extend the distributional Bochner formula [1, p. 72, Theorem 26] to certain kinds of distributions. Theorem I.1 gives a formula [Eq. (I.1.14)] which makes it possible to obtain easily the Fourier transform of distributions of the form As applications of the formula (I.1.14) we evaluate the Fourier transforms of the distributions G α ( P ± i 0, m , n ) [Eq. (I.4.1)] and H α ( P ± i 0,n) [Eq. (II.1.1)]. It follows from Theorem II.3 that H zk ( P ± i 0,n) is, for 2 K ≠ n +2 r , r =0,1..., an elementary solution of the n ‐dimensional ultrahyperbolic operator iterated k times.