### Hilbert Space Methods for Control Theoretic Splines: A Unified Treatment

M. Egerstedt, C. Martin, Y. Zhou
2006 Communications in Information and Systems
In this paper we give a basic derivation of smoothing and interpolating splines and through this derivation we show that the basic spline construction can be done through elementary Hilbert space techniques. Smoothing splines are shown to naturally separate into a filtering problem on the raw data and an interpolating spline construction. Both the filtering algorithm and the interpolating spline construction can be effectively implemented. We show that a variety of spline problems can be
more » ... ted into this common construction. By this construction we are also able to generalize the construction of smoothing splines to continuous data, a spline like filtering algorithm. Through the control theoretic approach it is natural to add multiple constraints and these techniques are developed in this paper. Introduction. Estimation and smoothing for data sets that contain deterministic and random data present difficulties not present in purely random data sets. Yet such data sets are very common in practice and if the nature of the data is not respected conclusions may be drawn that have little relation to reality. In this paper we will present a unified treatment of such problems. We will extend the theory of smoothing splines to cover such situations. Some of the techniques that we will use have been developed in papers by Egerstedt and Martin, [4, 8, 14] and their colleagues. The main technical contribution of this paper will be to show that many of these problems can be cast as minimum norm problems in suitable a Hilbert spaces. This approach unifies a series of problems that have been solved by Egerstedt, Zhou, Sun and Martin, [17, 18, 19] . Furthermore, the approach of this paper gives a unified treatment of smoothing splines as developed by Wahba, [15] , and the classical polynomial and exponential interpolating splines. The approach of this paper rests on the Hilbert space methods developed by Luenberger in [7] . The theory of smoothing splines is based on the premise that a datum, α is the sum of a deterministic part, β and a random part . It is assumed that is the value of a random variable from some probability distribution. Smoothing splines are designed to approximate the deterministic part by minimizing the variance of the random part. Often the random variable comes from measurement error. In the following examples the random error comes either from measurement or from estimation based on incomplete data.