Approximation of Integrals by Arithmetic Means and Related Matters
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).
KeywordsElectrostatic Field Quadrature Formula Minimal Potential Energy Faraday Cage Real Node
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