Some relations among RKHS norms, Fredholm equations, and innovations representations

We first show how reproducing kernel Hilbert space (RKHS) norms can be determined for a large class of covariance functions by methods based on the solution of a Riccati differential equation or a Wiener-Hopf integral equation. Efficient numerical algorithms for such equations have been extensively studied, especially in the control literature. The innovations representations enter in that it is they that suggest the form of the RKHS norms. From the RKHS norms, we show how recursive solutions can be obtained for certain Fredholm equations of the first kind that are widely used in certain approaches to detection theory. Our approach specifies a unique solution: moreover, the algorithms used are well suited to the treatment of increasing observation intervals.