Spline Functions and Stochastic Filtering

Abstract

Some relationships have been established between unbiased linear predictors of processes, in signal and noise models, minimizing the predictive mean square error and some smoothing spline functions. We construct a new family of multidimensional splines adapted to the prediction of locally homogeneous random fields, whose "$m$-spectral measure" (to be defined) is absolutely continuous with respect to Lebesgue measure and satisfies some minor assumptions. By considering partial splines, one may include an arbitrary drift in the signal. This type of correspondence underlines the potentialities of cross-fertilization between statistics and the numerical techniques in approximation theory.