We consider weak solutions of the zero-rest-mass (z.r.m.) equations described in Eastwood et al. (1981). The space of hyperfunctions, which contains the space of distributions, is defined and we consider hyper-function solutions of the equations on real Minkowski space $M^I$ and its conformal compactification $M$. We define a hyperfunction z.r.m. field to be future or past analytic if it is the boundary value of a holomorphic z.r.m. field on the future or past tube of complex Minkowski space respectively; and we demonstrate that any field on $M^I$ that is the sum of future and past analytic fields extends as a hyperfunction z.r.m. field to all of $M$. It is shown that any distribution solution on $M^I$ splits as required and hence extends as a hyperfunction solution to $M$. Twistor methods are then used to show that the same applies in the more general case of hyperfunction solutions on $M^I$. This leads to an alternative proof of the main result of Wells (1981): a hyperfunction z.r.m. field on compactified real Minkowski space is a unique sum of future and past analytic solutions.