Abstract We propose and analyze new numerical methods to evaluate fractional norms apply powers of elliptic operators. By means a reduced basis method, we project small dimensional subspace where explicit diagonalization via the eigensystem is feasible. The method relies on several independent evaluations $$({{\,\mathrm{I}\,}}-t_i^2\Delta )^{-1}f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Ma...