Statistical Optimality of Hermite Splines

Hermite splines are commonly used for interpolating data when samples of the derivative are available, in a scheme called Hermite interpolation. Assuming a suitable statistical model, we demonstrate that this method is actually optimal for reconstructing random signals in Papoulis' generalized sampling framework. We focus on second-order Lévy processes—the integrated version of Lévy processes—and rely on cubic Hermite splines to approximate the original continuous-time signal from its samples and its derivatives at integer values. We statistically justify the use of this reconstruction scheme by demonstrating the equivalence between cubic Hermite interpolation and the linear minimum mean-square error (LMMSE) estimation of a second-order Lévy process. We finally illustrate the cubic Hermite reconstruction scheme on an example of a discrete sequence sampled from the realization of a stochastic process.