Quadrature规则的谱极限与广义球面设计
Spectral Limitations of Quadrature Rules and Generalized Spherical Designs
摘要：
我们研究流形$M$配备了一个求积规则$$ \int_{M}{ϕ(x) dx} \simeq \sum_{i=1}^{n}{a_i ϕ(x_i)}.$$我们表明，$n-$点求积规则与非负的权重上的紧凑$d-$维流形不能集成超过最多的第一个$c_{d}n + o(n)$拉普拉斯特征函数准确。常数$c_d$是显式计算的，并且$c_2 = 4$。该结果是新的，甚至在$\mathbb{S}^2$，它概括了球形设计的结果。
英文摘要：
We study manifolds $M$ equipped with a quadrature rule $$ \int_{M}{ϕ(x) dx} \simeq \sum_{i=1}^{n}{a_i ϕ(x_i)}.$$ We show that $n-$point quadrature rules with nonnegative weights on a compact $d-$dimensional manifold cannot integrate more than at most the first $c_{d}n + o(n)$ Laplacian eigenfunctions exactly. The constants $c_d$ are explicitly computed and $c_2 = 4$. The result is new even on $\mathbb{S}^2$ where it generalizes results on spherical designs.
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