Orthogonal decompositions in Hilbertian subspaces, error functions and optimal extensions of interpolation systems (Q1610722)

The theory of variational splines and interpolation is closely connected with reproducing kernel Hilbert spaces. This paper discusses the extension of fixed subsets of domains to an interpolation system within such a reproducing kernel Hilbert space. The use of error functions, sometimes also called power functions, admits the introduction of an optimality criterion for these extensions. The connections of these extensions of interpolating systems to native spaces, Schur complements and (strictly) conditionally positive definite functions are especially emphasised. Several numerical examples are also given. The analysis of the spectrum of the interpolation matrix gives further links of this work to the practical aspects of interpolation with variational splines.