How fast do radial basis function interpolants of analytic functions converge?

The question in the title is answered using tools of potential theory. Convergence and divergence rates of interpolants of analytic functions on the unit interval are analysed. The starting point is a complex variable contour integral formula for the remainder in radial basis function (RBF) interpolation. We study a generalized Runge phenomenon and explore how the location of centres affects convergence. Special attention is given to Gaussian and inverse quadratic radial functions, but some of the results can be extended to other smooth basis functions. Among other things, we prove that, under mild conditions, inverse quadratic RBF interpolants of functions that are analytic inside the strip Im (z) < (1/2ε), where ε is the shape parameter, converge exponentially.