Discrete Interpolation Norms with Applications

We describe norm representations for interpolation spaces generated by finite-dimensional subspaces of Hilbert spaces. These norms are products of integer and noninteger powers of the Grammian matrices associated with the generating pair of spaces for the interpolation space. We include a brief description of some of the algorithms which allow the efficient computation of matrix powers. We consider in some detail the case of fractional Sobolev spaces both for positive and negative indices together with applications arising in preconditioning techniques. Numerical experiments are included.