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.
Résumé
La méthode concrète d'interpolation par surfaces-spline est étroitement liée au problème classique de la minimisation d'une semi-norme de Soboleff sous des contraintes d'interpolation; la structure intrinsèque des surfaces-spline est dès lors celle d'une extension multivariée des fonctions-spline naturelles. Le cadre abstrait adéquat est un espace fonctionnel hilbertien dont le noyau reproduisant ne fait pas intervenir de fonctions plus compliquées que des logarithmes et est aisé à programmer. Des formules commodes de représentation sont données, ainsi qu'une extension multivariée d'intérêt pratique du théorème du noyau de Peano. Grâce à la stabilité numérique de la factorisation de Cholesky des matrices symétriques définies positives, la construction d'une surface-spline peut se faire en exploitant point après point les données d'interpolation.
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Dedicated to Professor E. Stiefel
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Meinguet, J. Multivariate interpolation at arbitrary points made simple. Journal of Applied Mathematics and Physics (ZAMP) 30, 292–304 (1979). https://doi.org/10.1007/BF01601941
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DOI: https://doi.org/10.1007/BF01601941