An estimate for multivariate interpolation II
Journal of Approximation Theory
Bivariate Lagrange interpolation at the Padua points: The generating curve approach
Journal of Approximation Theory
A new class of oscillatory radial basis functions
Computers & Mathematics with Applications
Exact polynomial reproduction for oscillatory radial basis functions on infinite lattices
Computers & Mathematics with Applications
A Newton basis for Kernel spaces
Journal of Approximation Theory
Journal of Computational and Applied Mathematics
MMCS'08 Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces
Bases for conditionally positive definite kernels
Journal of Computational and Applied Mathematics
A new stable basis for radial basis function interpolation
Journal of Computational and Applied Mathematics
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It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered, the L 2 or L 驴驴驴 norms of interpolants can be bounded above by discrete 驴2 and 驴驴驴驴 norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample size and the fill distance.