Control Theoretic Smoothing Splines Are Approximate Linear Filters

The problem of constructing and approximating control theoretic smoothing splines is considered in this paper. It is shown that the optimal approximating function can be given as the solution of a forced Hamiltonian system, that can be explicitly solved using the Riccati transform, and an explicit linear filter can be constructed. We show that the bandwidth of the filter can be naturally controlled and thus for control theoretic smoothing splines the far past and the far future are unimportant. Hence smoothing splines are “local” in nature rather than “global”. We conclude that while spline approximations are not causal the far future is not important.