In approximation of functions based on point values, least-squares methods provide more stability than interpolation, at the expense of increasing the sampling budget. We show that near-optimal approximation error can nevertheless be achieved, in an expected L2 sense, as soon as the sample size m is larger than the dimension n of the approximation space by a constant ratio. On the other hand, for m = n, we obtain an interpolation strategy with a stability factor of order n.
This is joint work with Abdellah Chkifa.