Generalized Hermite interpolation via matrix-valued conditionally positive definite functions

Narcowich, Francis J.; Ward, Joseph D.

doi:10.1090/S0025-5718-1994-1254147-6

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Generalized Hermite interpolation via matrix-valued conditionally positive definite functions

Authors: Francis J. Narcowich and Joseph D. Ward
Journal: Math. Comp. 63 (1994), 661-687
MSC: Primary 41A05; Secondary 41A63
MathSciNet review: 1254147
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Abstract: In this paper, we consider a broad class of interpolation problems, for both scalar- and vector-valued multivariate functions subject to linear side conditions, such as being divergence-free, where the data are generated via integration against compactly supported distributions. We show that, by using certain families of matrix-valued conditionally positive definite functions, such interpolation problems are well poised; that is, the interpolation matrices are invertible. As a sample result, we show that a divergence-free vector field can be interpolated by a linear combination of convolutions of the data-generating distributions with a divergence-free, $3 \times 3$ matrix-valued conditionally positive definite function. In addition, we obtain norm estimates for inverses of interpolation matrices that arise in a class of multivariate Hermite interpolation problems.

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