Interpolation in the limit of increasingly flat radial basis functions

Many types of radial basis functions, such as multiquadrics, contain a free parameter. In the limit where the basis functions become increasingly flat, the linear system to solve becomes highly ill-conditioned, and the expansion coefficients diverge. Nevertheless, we find in this study that limiting interpolants often exist and take the form of polynomials. In the 1-D case, we prove that with simple conditions on the basis function, the interpolants converge to the Lagrange interpolating polynomial. Hence, differentiation of this limit is equivalent to the standard finite difference method. We also summarize some preliminary observations regarding the limit in 2-D.