Vector-valued Lg-splines. II: Smoothing splines (Q1063376)

In the first part of this series of papers [ibid. 70, 505-529 (1979; Zbl 0435.65007)] the authors have been concerned with the interpolation problem for vector-valued Lg-splines. In the present part, they continue their studies by considering the smoothing problem in the same vein. - For each integer \(k\geq 0\) denote by \(H_ k\) the real Sobolev space on a compact interval \([[ 0,T]]\) of the real line R. For fixed non-negative integers \(n_ 1,...,n_ p\) let H be the Cartesian product \(H=H_{n_ 1}\times H_{n_ 2}\times...\times H_{n_ p},\) then for a given \(p\times p\) matrix L of ordinary linear differential operators with ''regular'' coefficient functions, and arrays \((\lambda_ j)_{1\leq j\leq N}\) of continuous linear forms on H, real numbers \((r_ j)_{1\leq j\leq N}\), and strictly positive weights \((\rho_ j)_{1\leq j\leq N}\), a vector-valued smoothing Lg-spline is an element \(s\in H\) such that the quadratic sum functional \[ H\ni f\rightsquigarrow \int^{T}_{0}<Lf,Lf>d\tau +\sum_{1\leq j\leq N}\rho_ j(r_ j-<f,\lambda_ j>)^ 2\quad (T>0) \] is minimized over H. The existence and uniqueness of \(s\in H\) is an easy consequence of the orthoprojector method applied to the direct sum \(H\oplus R^ N\). The authors provide characterizations of the vector-valued smoothing Lg- splines by the reproducing kernel method via the matrix-valued Green's function of L. Moreover, the paper under review relates the vector-valued smoothing spline problem to a recursive problem of linear least squares estimation for a random process with observations in additive white noise. Finally, motivated by the theory of filtering, the authors describe a three-part algorithm which is applicable to the case when the continuous linear forms \((\lambda_ j)_{1\leq j\leq N}\) on H are defined as linear combinations of point evaluations of the derivatives of the coordinate functions associated with the element \(f\in H\).