A New Approach for the Existence Problem of Minimal Cubature Formulas Based on the Larman-Rogers-Seidel Theorem

A New Approach for the Existence Problem of Minimal Cubature Formulas Based on the Larman-Rogers-Seidel Theorem
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DOI：
10.1137/110826552
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发表时间：
2011-03
期刊：
SIAM J. Numer. Anal.
影响因子：
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通讯作者：
Masatake Hirao;Hiroshi Nozaki;M. Sawa;V. Vatchev
Masatake Hirao;Hiroshi Nozaki;M. Sawa;V. Vatchev
中科院分区：
其他
文献类型：
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作者：
Masatake Hirao;Hiroshi Nozaki;M. Sawa;V. Vatchev

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本文考虑球对称积分的4k +1次极小求积公式的存在性问题。我们证明了，对于一个极小公式在充分高维空间，存在一些同心球上的任何两个不同的点的内积是合理的。利用这一结果，我们证明了对任意d \ge2，不存在d维13次和21次的极小公式.
In this paper we consider the existence problem of minimal cubature formulas of degree $4k+1$ for spherically symmetric integrals. We prove that, for a minimal formula in sufficiently high-dimensional space, there exists some concentric sphere on which the inner product of any two distinct points is rational. By using this result, we prove that for any $d \ge 2$ there exist no $d$-dimensional minimal formulas of degrees $13$ and $21$ for some special integral.