We study Hilbert space aspects of the Klein-Gordon equation in
two-dimensional spacetime. We associate to its restriction to a spacelike wedge
a scattering from the past light cone to the future light cone, which is then
shown to be (essentially) the Hankel transform of order zero. We apply this to
give a novel proof, solely based on the causality of this spatio-temporal wave
propagation, of the theorem of de Branges and V. Rovnyak concerning Hankel
pairs with a support property. We recover their isometric expansion as an
application of Riemann's general method for solving Cauchy-Goursat problems of
hyperbolic type.