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Multivariate Approximation Theory pp 359–377Cite as

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Construction of Cubature Formulae Using Real Ideals

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Abstract

In this paper a new approach is outlined to construct cubature formulae for two-dimensional integrals. The method which will be presented is based on a fundamental connection between real ideals and cubature formulae. The corresponding theorem is proved in [8]. This basic theorem and some necessary items will be studied in the first section. The construction of a class of cubature formulae will be derived in the next section. Finally some applications will be given.

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References

  1. Dubois, D.W. and G. Efroymson: Algebraic theory of real varieties. I. Studies and essays presented to Yu-Why Chen on his sixtieth birthday. October 1970. 107–135.

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  5. Risler, J.J.: Une characterisation des idéaux des variétés algébriques réelles. Note aux CRAS. Paris 21, 1171–1173 (1970).

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  6. Schmid, H.J.: On Cubature Formulae with a minimal number of knots. Numer. Math. 31, 282–297 (1978).

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  7. Schmid, H.J.: On Gaussian cubature formulae of degree 2k-1. To appear in “Numerical Integration”, Oberwolfach 1978, Ed. G. Hämmerlin.

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  8. Schmid, H.J.: Interpolatorische Kubaturformeln und reelle Ideale. To appear.

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  9. Stroud, A.H.: Approximate calculation of multiple integrals. Englewood Cliffs, New Jersey: Prentice Hall 1971.

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Authors
  1. Hans Joachim Schmid
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Editors and Affiliations

  1. Lehrstuhl für Mathematik I, Universität Siegen, Hölderlinstraße 3, D-5900, Siegen 21, Western Germany

    Walter Schempp

  2. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-7400, Tübingen 1, Western Germany

    Karl Zeller

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© 1979 Springer Basel AG

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Schmid, H.J. (1979). Construction of Cubature Formulae Using Real Ideals. In: Schempp, W., Zeller, K. (eds) Multivariate Approximation Theory. ISNM International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 51. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6289-9_24

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  • DOI: https://doi.org/10.1007/978-3-0348-6289-9_24

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-1102-5

  • Online ISBN: 978-3-0348-6289-9

  • eBook Packages: Springer Book Archive

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