Optimal methods of interpolation in Nonparametric Regression

Abstract

Within the framework of Optimal Recovery, optimal methods of interpolation, based on the Abel–Jacobi elliptic functions, have been found for some Hardy classes of analytic functions [9]. It will be shown that these methods are also optimal according to criteria of Optimal Design and Nonparametric Regression.

For all noise levels away from 0, the mean squared error of the optimal interpolant is evaluated explicitly, in a non-asymptotic setting. In this result, a pivotal role is played by an interference effect in which both stochastic and deterministic parts of the interpolant exhibit an oscillating behavior, with the two oscillating functions canceling each other.