  In a previous paper [Adcock & Huybrechs, 2019] we described the numerical
approximation of functions using redundant sets and frames. Redundancy in the
function representation offers enormous flexibility compared to using a basis,
but ill-conditioning often prevents the numerical computation of best
approximations. We showed that, in spite of said ill-conditioning,
approximations with regularization may still provide accuracy up to order
$\sqrt{\epsilon}$, where $\epsilon$ is a small truncation threshold. When using
frames, i.e. complete systems that are generally redundant but which provide
infinite representations with coefficients of bounded norm, this accuracy can
actually be achieved for all functions in a space. Here, we generalize that
setting in two ways. We assume information or samples from $f$ from a wide
class of linear operators acting on $f$, rather than inner products associated
with the best approximation projection. This enables the analysis of fully
discrete approximations based, for instance, on function values only. Next, we
allow oversampling, leading to least-squares approximations. We show that this
leads to much improved accuracy on the order of $\epsilon$ rather than
$\sqrt{\epsilon}$. Overall, we demonstrate that numerical function
approximation using redundant representations may lead to highly accurate
approximations in spite of having to solve ill-conditioned systems of
equations.
