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Upper and lower estimates for numerical integration errors on spheres of arbitrary dimension
Classical Analysis and ODEs · Submitted 16 Jan 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For these spaces we obtain similar upper and lower bounds for the worst case integration error.
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The abstract, sentence by sentence
In this paper we study the worst-case error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain spaces of continuous functions on $\mathbb{S}^{d}$. For the classical Sobolev spaces $\mathbb{H}^s(\mathbb{S}^d)$ ($s>\frac d2$) upper and lower bounds for the worst case integration error have been obtained By Brauchart, Hesse, and Sloan earlier in papers. We investigate the behaviour for $s\to\frac d2$ by introducing spaces $\mathbb{H}^{\frac d2,\gamma}(\mathbb{S}^d)$ with an extra logarithmic weight. For these spaces we obtain similar upper and lower bounds for the worst case integration error.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this paper we study the worst-case error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain spaces of continuous functions on $\mathbb{S}^{d}$. | contribution 0.65 | 0.61 | 0.04 | asserted 0.88 |
| s1 | For the classical Sobolev spaces $\mathbb{H}^s(\mathbb{S}^d)$ ($s>\frac d2$) upper and lower bounds for the worst case integration error have been obtained By Brauchart, Hesse, and Sloan earlier in papers. | background 1.00 | 0.03 | 0.06 | asserted 1.00 |
| s2 | We investigate the behaviour for $s\to\frac d2$ by introducing spaces $\mathbb{H}^{\frac d2,\gamma}(\mathbb{S}^d)$ with an extra logarithmic weight. | contribution 0.84 | 0.91 | 0.08 | asserted 0.98 |
| s3 | For these spaces we obtain similar upper and lower bounds for the worst case integration error. | result 0.99 | 0.94 | 0.64 | asserted 1.00 |
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The paper
Original abstract
In this paper we study the worst-case error of numerical integration on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain spaces of continuous functions on $\mathbb{S}^{d}$. For the classical Sobolev spaces $\mathbb{H}^s(\mathbb{S}^d)$ ($s>\frac d2$) upper and lower bounds for the worst case integration error have been obtained By Brauchart, Hesse, and Sloan earlier in papers. We investigate the behaviour for $s\to\frac d2$ by introducing spaces $\mathbb{H}^{\frac d2,\gamma}(\mathbb{S}^d)$ with an extra logarithmic weight. For these spaces we obtain similar upper and lower bounds for the worst case integration error.
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How the literature uses it
1 arXiv paper cite this in 2 sentences
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Let the space $ℍ^(d/2,γ)(𝕊^d)$, $γ>12$, (see [ref] ) be the set of all functions $f∈𝕃_2(𝕊^d)$ with finite norm [equation]
Comparison of probabilistic and deterministic point sets · Estimates for energy integrals in the nonsingular case · 0.87
What it cites · 1 sentence without a confident label
Uses the method or tool of 1 paper
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Dai [ref] , which we will refer to in more detail in Section [ref].
Geodesic distance Riesz energy on the sphere · Introduction · 0.90
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