Approximation of Integrals by Arithmetic Means and Related Matters

  • Jacob Korevaar
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


This is a survey of recent work by a small group investigating Chebyshev-type quadratures for large numbers of nodes and related potential theory. Besides the author, the participants were A.B.J. Kuijlaars, J.L.H. Meyers and M.A. Monterie. Much of the work concerns nice “surfaces” in R d, d ≥ 1, equipped with normalized “area” measure. Fundamental results of S.N. Bernstein for the interval [-1, 1] are extended and applied. It is shown that minimum-norm formulas exhibit massive coalescence of nodes. Other results involve domains of product type including the sphere. On the sphere, good N-tuples of nodes correspond to configurations of N point charges 1/N for which the electrostatic field is very small on the compact subsets of the unit ball (“Faraday cage effect”). By “Several complex variables” it becomes plausible that this field can be made as small as exp(-cN 1/2) in the case of S 2. This observation supports our conjecture that there exist N-tuples of distinct nodes on S 2 which give Chebyshev-type formulas that are polynomially exact to degree p ~ cN 1/2 (so that there are “spherical p-designs” consisting of N = O(p2) points).


Electrostatic Field Quadrature Formula Minimal Potential Energy Faraday Cage Real Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • Jacob Korevaar
    • 1
  1. 1.Faculty of Mathematics and Computer ScienceUniversity of AmsterdamAmsterdamThe Netherlands

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