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Cubature Formula

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Euclidean Design Theory

Part of the book series: SpringerBriefs in Statistics ((JSSRES))

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Abstract

A cubature formula reveals a numerical integration rule that approximates a multiple integral by a positive linear combination of function values at finitely many specified points on the integral domain. A central objective is to investigate the existence as well as the construction of cubature formulas in high dimensions.

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  • 22 January 2020

    In the original version of the book, the following belated corrections have been made

Notes

  1. 1.

    Let \(\mathscr {B}\) be the Borel \(\sigma \)-algebra of \(\varOmega \). Denote the support of \(\mu \) by \(\mathop {\mathrm {supp}}\nolimits (\mu )\), i.e., closure of \(\{ B \in \mathscr {B}\mid \mu (B) > 0 \}\).

  2. 2.

    An odd polynomial f is a polynomial satisfying \(f(-\omega ) = -f(\omega )\) for all \(\omega \in \varOmega \).

  3. 3.

    See [39, Theorems 12.1,12.2] for details of the proofs of Theorems 2.4 and 2.5.

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Authors and Affiliations

  1. Graduate School of System Informatics, Kobe University, Kobe, Hyogo, Japan

    Masanori Sawa

  2. School of Information and Science Technology, Aichi Prefectural University, Nagakute, Aichi, Japan

    Masatake Hirao

  3. Research Center for Mathematics and Science Education, Tokyo University of Science, Tokyo, Japan

    Sanpei Kageyama

  4. Hiroshima University, Hiroshima, Japan

    Sanpei Kageyama

Authors
  1. Masanori Sawa
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  2. Masatake Hirao
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  3. Sanpei Kageyama
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Correspondence to Masanori Sawa .

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Sawa, M., Hirao, M., Kageyama, S. (2019). Cubature Formula. In: Euclidean Design Theory. SpringerBriefs in Statistics(). Springer, Singapore. https://doi.org/10.1007/978-981-13-8075-4_2

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