  We study the approximation of a square-integrable function from a finite
number of evaluations on a random set of nodes according to a well-chosen
distribution. This is particularly relevant when the function is assumed to
belong to a reproducing kernel Hilbert space (RKHS). This work proposes to
combine several natural finite-dimensional approximations based two possible
probability distributions of nodes. These distributions are related to
determinantal point processes, and use the kernel of the RKHS to favor
RKHS-adapted regularity in the random design. While previous work on
determinantal sampling relied on the RKHS norm, we prove mean-square guarantees
in $L^2$ norm. We show that determinantal point processes and mixtures thereof
can yield fast convergence rates. Our results also shed light on how the rate
changes as more smoothness is assumed, a phenomenon known as superconvergence.
Besides, determinantal sampling generalizes i.i.d. sampling from the
Christoffel function which is standard in the literature. More importantly,
determinantal sampling guarantees the so-called instance optimality property
for a smaller number of function evaluations than i.i.d. sampling.
