H-sets for kernel-based spaces

The concept of H-sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and H-sets exist, but are only rarely accessible by mathematical...

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Bibliographic Details
Published in:Journal of approximation theory 2023-10, Vol.294, p.105942, Article 105942
Main Author: Schaback, Robert
Format: Article
Language:English
Subjects:
Alternation
Approximation
Duality
Error bounds
Kernels
Radial basis functions
Reproducing kernel Hilbert spaces
Stability
Uniqueness
Citations: Items that this one cites
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Summary:The concept of H-sets as introduced by Collatz in 1956 was very useful in univariate Chebyshev approximation by polynomials or Chebyshev spaces. In the multivariate setting, the situation is much worse, because there is no alternation, and H-sets exist, but are only rarely accessible by mathematical arguments. However, in Reproducing Kernel Hilbert spaces, H-sets are shown here to have a rather simple and complete characterization. As a byproduct, the strong connection of H-sets to Linear Programming is studied. But on the downside, it is explained why H-sets have a very limited range of applicability in the times of large-scale computing.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2023.105942