A generalization of the Poisson transform

Abstract

When limits are omitted from an integral appearing in the text the range of integration is understood to be (cc, cc). At the conclusion of the paper we shall give a complex inversion formula for (2). The preceding transforms include as special cases the Poisson transform (A=O, B=C=1) recently studied by Pollard [1], and the conjugate of the Poisson transform (B =0, A = C= 1) whose kernel is the Hilbert transform of the Poisson kernel. The inversion of (1) and (2) is intended as a first step toward the solution of the problem of inverting the convolution transform whose kernel is a rational function with no poles on the real axis. I am indebted to Professor Pollard for proposing the problem considered in this paper. For the heuristic motivation of our inversion formulas we refer to [1 ] and [2 ]. We distinguish two cases according as A is or is not zero. In the former case a slight modification of Pollard's proof for the Poisson case shows that (1) is, inverted by