Analysis and design of minimax-optimal interpolators

We consider a class of interpolation algorithms, including the least-squares optimal Yen (1956) interpolator, and we derive a closed-form expression for the interpolation error for interpolators of this type. The error depends on the eigenvalue distribution of a matrix that is specified for each set of sampling points. The error expression can be used to prove that the Yen interpolator is optimal. The implementation of the Yen algorithm suffers from numerical ill conditioning, forcing the use of a regularized, approximate solution. We suggest a new, approximate solution consisting of a sinc-kernel interpolator with specially chosen weighting coefficients. The newly designed sinc-kernel interpolator is compared with the usual sinc interpolator using Jacobian (area) weighting through numerical simulations. We show that the sinc interpolator with Jacobian weighting works well only when the sampling is nearly uniform. The newly designed sinc-kernel interpolator is shown to perform better than the sinc interpolator with Jacobian weighting.