In the 1990s Kenji Yajima carried out a comprehensive analysis of the L^p boundedness of the classical wave operators from scattering theory. In recent joint work with Marius Beceanu, we obtained a representation of the wave operators in R^3 as a superposition of translations and reflections. This work combines elements of Yajima’s work with methods from harmonic analysis. Specifically, we rely on both Stein-Tomas restriction theory of the Fourier transform, and Wiener’s theorem on inversion in a convolution algebra, albeit a complicated one. The latter is essentially a summation method and is used to sum a Born series with large terms. The necessary condition for invertibility needed in Wiener inversion comes from spectral theory developed about 13 years ago. In effect, this amounts to the classical Agmon-Kato-Kuroda theory but completely redone by means of Stein-Tomas type results.