2021/07/21 by Robert Schaback, Schaback, Robert
Mathematics · Physics and Astronomy · #41A10 #41A52 #65D15 #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical Analysis (math.NA) #Statistical Mechanics and Entropy #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.2107.10040
openalex publication_date 2021/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.