Abstract
The concrete method of ‘surface spline interpolation’ is closely connected with the classical problem of minimizing a Sobolev seminorm under interpolatory constraints; the intrinsic structure of surface splines is accordingly that of a multivariate extension of natural splines. The proper abstract setting is a Hilbert function space whose reproducing kernel involves no functions more complicated than logarithms and is easily coded. Convenient representation formulas are given, as also a practical multivariate extension of the Peano kernel theorem. Owing to the numerical stability of Cholesky factorization of positive definite symmetric matrices, the whole construction process of a surface spline can be described as a recursive algorithm, the data relative to the various interpolation points being exploited in sequence.
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