Observations on the behavior of radial basis function approximations near boundaries

RBF approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of PDEs. RBFs allow arbitrarily scattered data, generalize easily to several space dimensions, and can be spectrally accurate. However, accuracy degradations near boundaries in many cases severely limit the utility of this approach. With that as motivation, this study aims at gaining a better understanding of the properties of RBF approximations near the ends of an interval in 1-D and towards edges in 2-D.