Abstract
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 ∞ norms of interpolants can be bounded above by discrete ℓ2 and ℓ ∞ 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.
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Communicated by Joe Ward.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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De Marchi, S., Schaback, R. Stability of kernel-based interpolation. Adv Comput Math 32, 155–161 (2010). https://doi.org/10.1007/s10444-008-9093-4
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DOI: https://doi.org/10.1007/s10444-008-9093-4