A Rational Interpolation Scheme with Superpolynomial Rate of Convergence

The purpose of this study is to construct a high-order interpolation and scheme for arbitrary scattered datasets. The resulting function approximation is an interpolation function when the dataset is exact, or a regression if measurement errors are present. We represent each datapoint with a Taylor series, and the approximation error as a combination of the derivatives of the the target function. A weighted sum of the square of the coefficient of each derivative term in the approximation error is minimized to obtain the interpolation approximation. The resulting approximation function is a high-order rational function with no poles. When measurement errors are absent, the interpolation approximation converges to the target function faster than any polynomial rate of convergence.