Vector Splines and the Estimation of Filter Functions

In this paper, we consider the estimation of vector-valued periodic functions by use of splines. An error criterion is constructed and it is shown that the vector spline is simply the vector of univariate splines. The vector-valued spline is shown to be an approach to the estimation of the spectral density matrix. In a multivariate setting, two approaches are given for estimation of filter-related functions, including gain, transfer function, coherency, and so on. Some results on consistency are given and an application is made to sonar systems.