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Abstract

For integrals with convex domains of integration we consider cubature formulae of degree r, r∈{2, 3}, with only positive weights and all nodes inside the domain. We show, that the minimal number of nodes for such formulae varies from 3 to at least 5 (for r=2) and from 3 to at least 9 (for r=3) in dependence of the shape of the domain.

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References

  1. [1]
    FRITSCH, F.N.: On the number of nodes in self-contained integration formulas of degree two for compact planar regions. Numer. Math. 16, 224–230 (1970).CrossRefGoogle Scholar
  2. [2]
    GÜNTHER, C.: Über die Anzahl von Stützstellen in mehrdimensionalen Integrationsformein mit positiven Gewichten, thesis Univ. Karlsruhe, 1973.Google Scholar
  3. [3]
    GÜNTHER, C.: Nichtnegative Polynome in zwei Veränderlichen. J. Approx. theory 15, 124–131 (1975).CrossRefGoogle Scholar
  4. [4]
    KARLIN, S., and STUDDEN, W.J.: Tchebycheff Systems, Intersience Publisher Inc. New York 1966.Google Scholar
  5. [5]
    MÖLLER, H.M.: Lower bounds for the number of nodes in cubature formulae, in “Numerische Integration” (ed. G. Hämmerlin) ISNM 45, 221–230, Birkhäuser Verlag 1979.CrossRefGoogle Scholar
  6. [6]
    MYSOVSKIH, I.P.: On Chakalov’s theorem. USSR Comp. Math. 15, 221–227, 1975.CrossRefGoogle Scholar
  7. [7]
    SCHMID, H.J.: Interpolatorische Kubaturformeln, Habilitationsschrift, Univ. Erlangen 1980.Google Scholar
  8. [8]
    TCHAKALOFF, V.: Formules de cubature mécaniques à coefficients non-négatifs, Bull. Sci. Math. 12, 123–134, 1957.Google Scholar

Copyright information

© Springer Basel AG 1982

Authors and Affiliations

  • H. M. Möller

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