Control Theoretic Splines with Deterministic and Random Data

In this paper we give a basic derivation of smoothing splines and through this derivation we show that the basic smoothing spline construction can be separated into a Altering problem on the raw data and an interpolating spline construction. Both the filtering algorithm and and the interpolating spline construction can be effectively implemented. We allow hard constraints (such as boundary values) on the dynamics and we allow data that is subject to error. We are thus constructing smoothing splines with hard constraints.