On Some Classes of Solutions of the Wave Equation ∂t2f−Δf=0

Abstract

We investigate certain classes of solutions of the wave equation for which Rf(x) = −(1/x 2)f(+x/x 2) is a well‐defined C ∞ solution for all points of the Minkowski space, if f(x) is a C ∞ solution. We mainly exploit the fact that the transformation R is a generalized Hankel transformation in momentum space. We make use of several recent results obtained by Zemanian in connection with the Hankel transformation and construct a self‐adjoint representation of the Lie algebra of the group O(2, 4) which is an invariance group of the wave equation. Finally, we construct and discuss the eigenfunctions of R in Minkowski space.