Optimal Interpolation in Semi-Hilbert Spaces

Tobias Werther

NuHAG, Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria.

Polynomial splines can be introduced from three different viewpoints. They can be regarded as solutions of an extremal problem in the context of interpolation, or simply defined as piecewise polynomials with global smoothness properties, or they appear as linear combinations of certain basis splines, so-called B-splines. All these concepts are equivalent in the univariate case. The tensor product of univariate splines is a straight forward way to define multivariate splines. However, the resulting grid structure is quite restrictive and only suitable for rather simple domains. Besides the tensor construct, one can take any of the three mentioned viewpoints for extending splines to multivariate structures. Unfortunately, it turns out that the multivariate case is completely different from the univariate case in the sense that there is no longer any unified framework covering the various extensions. Indeed, looking for an optimal interpolant in the sense of smoothness does, in general, not lead to piecewise polynomials. This phenomenon is mainly due to the fact that there is no longer any total ordering of nodes and that, unlike the univariate case, there are simply too many ways of partitioning a domain into cells.

In this work we are mainly focusing on the extension following the lines of the extremal problem in the context of interpolation. It retains most of the physical aspects of the spline model and gives rise to the notion of thin plate splines. This technique has been originally studied by J. Duchon . It leads directly to conditionally positive definite functions and associated native spaces. Conditionally positive definite functions, in particular, radial basic functions, and associated native Hilbert spaces have been extensively studied for the case where the native Hilbert space consists of continuous functions. In this work we extend both concepts to what we call semi-Hilbert kernel and semi-Hilbert space in the framework of locally convex vector spaces for which the well-known cases are special examples. We discuss the relation between semi-Hilbert kernels and semi-Hilbert spaces. We further study optimal interpolation and state necessary and sufficient conditions for unique solutions. This covers many interpolation problems encountered in practice, in particular, optimal recovery from generalized Hermite-Birkhoff data. We further analyze a special class of kernel functions including all cases that have been considered of practical interest so far. We then present a numerical scheme for approximating the optimal interpolation using tensor products of multivariate B-splines.

An interesting aspect of radial basic function interpolation is the fact that the arising system of linear equations is similar to the one that comes from best linear unbiased estimation in stochastic data modelling. Best linear unbiased estimation is also referred to as kriging in the geo-science community. If we use the dual formulation of kriging, we recover the same equations that result from radial basic functions. This common point, known for some time, has been reported by Matheron in 1981. Due to the semi-Hilbert kernel theory, we provide a rigorous description of the equivalence between optimal interpolation and optimal estimation.

Altogether, it turns out that the definition of the multivariate spline-model in a variational framework and the extension by means of B-splines are both of high relevance though not equivalent. The first is important for capturing physical properties, the latter for numerical simulations.

Keywords: Optimal interpolation, semi-Hilbert spaces, semi-Hilbert kernels, conditionally positive definite functions, splines, radial basic functions, quadratic programming, krigging,.