  We establish an equivalence between two classes of methods for solving
fractional diffusion problems, namely, Reduced Basis Methods (RBM) and Rational
Krylov Methods (RKM). In particular, we demonstrate that several recently
proposed RBMs for fractional diffusion can be interpreted as RKMs. This changed
point of view allows us to give convergence proofs for some methods where none
were previously available.
  We also propose a new RKM for fractional diffusion problems with poles chosen
using the best rational approximation of the function $x^{-s}$ in the spectral
interval of the spatial discretization matrix. We prove convergence rates for
this method and demonstrate numerically that it is competitive with or superior
to many methods from the reduced basis, rational Krylov, and direct rational
approximation classes. We provide numerical tests for some elliptic fractional
diffusion model problems.
