Cubature formulas in numerical analysis and Euclidean tight designs
Cubature formulas in numerical analysis and Euclidean tight designs
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DOI：
10.1016/j.ejc.2009.03.035
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发表时间：
2010-02
期刊：
影响因子：
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通讯作者：
E. Bannai;E. Bannai;Masatake Hirao;M. Sawa
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文献类型：
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作者：
E. Bannai;E. Bannai;Masatake Hirao;M. Sawa
在组合学中，欧几里德t-设计的概念首先由Neumaier-Seidel（1988）[25]定义，作为球面t-设计概念的两步推广。在分析中可以把欧氏t-设计看作一般容积公式的一种特殊情况。我们指出，工作容积公式的穆勒和其他人（这不是很清楚的组合），是非常重要的研究欧几里德t-设计。特别是，他们澄清了什么是紧欧几里德t-设计的正确定义（紧t-设计上的R-紧t-设计p-同心球）的问题。因此，本文的第一个目的是告诉组合数学家，关于体积公式的理论在分析中的重要性。同时，我们认为将我们对欧几里德t-设计的观点传达给分析者是很重要的。本文的第二个目的是回顾紧欧几里德t-设计的研究进展。有许多新的有趣的例子和丰富的理论紧欧几里德t-设计。本文详细讨论了R2中的紧欧氏t-设计，并讨论了紧欧氏t-设计研究的下一步工作。此外，我们还研究了紧欧几里得t-设计的已知例子与高斯t-设计的对应关系。
In combinatorics, the concept of Euclidean t-design was first defined by Neumaier–Seidel (1988) [25], as a two-step generalization of the concept of spherical t-design. It is possible to regard Euclidean t-design as a special case of general cubature formulas in analysis. We point out that the works on cubature formulas by Möller and others (which were not well aware by combinatorialists), are very important for the study of Euclidean t-designs. In particular, they clarify the question of what is the right definition of tight Euclidean t-designs (tight t-designs on Rnand tight t-designs on p-concentric sphere). So, the first purpose of this paper is to tell combinatorialists, the importance of the theory on cubature formulas in analysis. At the same time we think that it is important for us to communicate our viewpoint of Euclidean t-designs to the analysts. The second purpose of this paper is to review the developments of the research on tight Euclidean t-designs. There are many new interesting examples and rich theories on tight Euclidean t-designs. We discuss the tight Euclidean t-designs in R2carefully, and we discuss what will be the next stage of the study on tight Euclidean t-designs. Also, we investigate the correspondence of the known examples of tight Euclidean t-designs with the Gaussian t-designs.