Abstract
We provide a symmetric preconditioning method based on weighted divided differences which can be applied in order to solve certain ill-conditioned scattered-data interpolation problems in a stable way; more precisely, our method applies to cases where the ill-conditioning comes from the fact that the basis function is slowly growing, and the number of interpolation points is large. Concerning the theoretical background, an a priori unbounded operator on ℓ2(ℤ) is preconditioned so as to get a bounded and coercive operator. The method has another and probably even more interesting interpretation in terms of constructing certain Riesz bases of appropriate closed subspaces ofL 2(ℝ). In extending Mallat’s multiresolution analysis to the scattered data case, we construct nested sequences of spaces giving rise to orthogonal decompositions of functions inL 2(ℝ); in this way the idea of wavelet decompositions is (theoretically) carried over to scattered-data methods.
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Jetter, K., Stöckler, J. A generalization of de Boor’s stability result and symmetric preconditioning. Adv Comput Math 3, 353–367 (1995). https://doi.org/10.1007/BF03028368
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DOI: https://doi.org/10.1007/BF03028368