交换扩展与求积公式
Commuting Extensions and Cubature Formulae
AI总结：
基于一维高斯求积的新观点，提出通过构造交换矩阵扩展来计算d维求积公式的方法，并证明求积公式与满足特定相容性条件的交换扩展之间的对应关系。
摘要：
基于一维高斯求积的新观点，我们提出了一种计算d维求积公式的新方法。众所周知，一维高斯求积的节点可以通过所谓的Jacobi矩阵的特征值计算。d维类比是，求积节点可以从某些相互交换的矩阵的特征值获得。这些矩阵是通过扩展（添加行和列）某些与R^d中的坐标算子x_1,...,x_d相关的非交换矩阵A_1,...,A_d得到的。我们证明了求积公式与A_1,...,A_d的“交换扩展”之间的对应关系，这些扩展满足一个相容性条件，在适当的坐标下，该条件约束扩展矩阵中的某些块为零。因此，寻找求积公式的问题可以转化为计算（然后同时对角化）交换扩展的问题。我们讨论了交换扩展的存在性和预期大小，并描述了计算它们的尝试，以及使用新方法得到的求积公式的例子。
英文摘要：
Based on a novel point of view on 1-dimensional Gaussian quadrature, we present a new approach to the computation of d-dimensional cubature formulae. It is well known that the nodes of 1-dimensional Gaussian quadrature can be computed as eigenvalues of the so-called Jacobi matrix. The d-dimensional analog is that cubature nodes can be obtained from the eigenvalues of certain mutually commuting matrices. These are obtained by extending (adding rows and columns to) certain noncommuting matrices A_1,...,A_d, related to the coordinate operators x_1,...,x_d, in R^d. We prove a correspondence between cubature formulae and "commuting extensions" of A_1,...,A_d, satisfying a compatibility condition which, in appropriate coordinates, constrains certain blocks in the extended matrices to be zero. Thus the problem of finding cubature formulae can be transformed to the problem of computing (and then simultaneously diagonalizing) commuting extensions. We give a general discussion of existence and of the expected size of commuting extensions and describe our attempts at computing them, as well as examples of cubature formulae obtained using the new approach.
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