The problem considered in this paper is the continuous-energy, continuous-space time-independent neutron-diffusion equation, with given source and zero flux at the boundary. The basic result is that Galerkin-type spectral synthesis approximations converge optimally to the exact solution as the number of trial spectra increases, provided the diffusion coefficient and total macroscopic cross section are spatially homogeneous, and other (more) reasonable conditions of a technical nature are satisfied. The proof makes use of the general results of Pol'skii, which give sufficient conditions for the convergence of any projection method using the same trial and test spaces. As an application of the basic result, it is shown that the classic multigroup method converges optimally provided the maximum group width over any fixed bounded energy interval approaches zero. Several directions are indicated for possible related future work.