  We propose and analyze new numerical methods to evaluate fractional norms and
apply fractional powers of elliptic operators. By means of a reduced basis
method, we project to a small dimensional subspace where explicit
diagonalization via the eigensystem is feasible. The method relies on several
independent evaluations of $(\textrm{I}-t_i^2\Delta)^{-1}f$, which can be
computed in parallel. We prove exponential convergence rates for the optimal
choice of sampling points $t_i$, provided by the so-called Zolotar\"ev points.
Numerical experiments confirm the analysis and demonstrate the efficiency of
our algorithm.
