重心插值可以说是数值多项式插值的首选方法。多项式插值使用所谓的重心权重以函数值表示，这些权重依赖于插值点。已知的重心权重显式公式很少。在[H. Wang and S. Xiang, Math. Comp., 81 (2012), 861--877]中，作者证明了Legendre多项式根的重心权重可以用相应高斯求积规则的权重显式表示。该思想随后在Chebfun包[L. N. Trefethen and others, The Chebfun Development Team, 2011]中实现，并在过程中由Chebfun作者推广到Jacobi、Laguerre和Hermite多项式的根。在本文中，我们探讨了重心权重与高斯求积之间联系的普遍性，并表明这种关系与正交多项式的降算子的存在有关。我们提供了详尽的案例列表，其中恢复了所有已知公式，并推导了一些新公式，包括Gauss-Radau和Gauss-Lobatto点的重心权重。基于Hale和Townsend提出的快速${\mathcal O}(n)$高斯求积算法，这导致了重心权重的${\mathcal O}(n)$计算方案。
英文摘要：
Barycentric interpolation is arguably the method of choice for numerical polynomial interpolation. The polynomial interpolant is expressed in terms of function values using the so-called barycentric weights, which depend on the interpolation points. Few explicit formulae for these barycentric weights are known. In [H. Wang and S. Xiang, Math. Comp., 81 (2012), 861--877], the authors have shown that the barycentric weights of the roots of Legendre polynomials can be expressed explicitly in terms of the weights of the corresponding Gaussian quadrature rule. This idea was subsequently implemented in the Chebfun package [L. N. Trefethen and others, The Chebfun Development Team, 2011] and in the process generalized by the Chebfun authors to the roots of Jacobi, Laguerre and Hermite polynomials. In this paper, we explore the generality of the link between barycentric weights and Gaussian quadrature and show that such relationships are related to the existence of lowering operators for orthogonal polynomials. We supply an exhaustive list of cases, in which all known formulae are recovered and also some new formulae are derived, including the barycentric weights for Gauss-Radau and Gauss-Lobatto points. Based on a fast ${\mathcal O}(n)$ algorithm for the computation of Gaussian quadrature, due to Hale and Townsend, this leads to an ${\mathcal O}(n)$ computational scheme for barycentric weights.
BWLer: Barycentric Weight Layer Elucidates a Precision-Conditioning Tradeoff for PINNs
Institute for Computational & Mathematical Engineering, Stanford University（计算与数学工程研究所，斯坦福大学）；Department of Bioengineering, Stanford University（生物工程系，斯坦福大学）；Department of Computer Science, Stanford University（计算机科学系，斯坦福大学）；Department of Computer Science & Engineering, University at Buffalo（计算机科学与工程系，布法罗大学）
