The approximation of Coulomb continuum functions by an L 2 basis is studied using a Laguerre� function basis which can be extended to completeness. Also studied is the convergence rate of L2 approximations to Born matrix elements for electron impact ionisation as a function of basis�set size. This important class of matrix elements occurs in pseudo�state close-coupling calculations, accounting for scattering to the three�body continuum. Convergence rates in both cases are derived analytically and confirmed numerically. We find that the rate of pointwise convergence of L2 expansions to the continuum function is slow, and of conditional type; however, it is proven that the corresponding ionisation matrix elements converge geometrically, Our result agrees with the behaviour observed in pseudo�state calculations.