Spectral Limitations of Quadrature Rules and Generalized Spherical Designs
Abstract We study manifolds $M$ equipped with a quadrature rule \begin{equation*} \int_{M}{\phi(x)\,\mathrm{d}x} \simeq \sum_{i=1}^{n}{a_i \phi(x_i)}.\end{equation*}We show that $n$-point quadrature rules with nonnegative weights on a compact $d$-dimensional manifold cannot integrate more than at most the 1st $c_{d}n + o(n)$ 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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2015 ◽
Vol 34e
(1and2)
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pp. 61
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2012 ◽
Vol 2
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pp. 10-15
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2020 ◽
Vol 367
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pp. 124769
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2016 ◽
Vol 28
(5)
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pp. 278-285
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