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Quadrature in Besov spaces on the Euclidean sphere
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posted on 2023-06-08, 09:38 authored by K Hesse, H N Mhaskar, I H SloanLet $q\\geq 1$ be an integer, $\\mathbb{S}^q$ denote the unit sphere embedded in the Euclidean space $\\mathbb{R}^{q+1}$, and $\\mu_q$ be its Lebesgue surface measure. We establish upper and lower bounds for \\[ \\sup_{f\\in {\\cal B}^\\gamma_{p,\\rho}} \\left|\\int_{\\mathbb{S}^q} f d\\mu_q - \\sum_{k=1}^M w_k f(\\mathbf{x}_k)\\right|, \\qquad \\mathbf{x}_k\\in\\mathbb{S}^q,\\ w_k\\in\\mathbb{R},\\ k=1,\\cdots,M, \\] where ${\\cal B}^\\gamma_{p,\\rho}$ is the unit ball of a suitable Besov space on the sphere. The upper bounds are obtained for choices of $\\mathbf{x}_k$ and $w_k$ that admit exact quadrature for spherical polynomials of a given degree, and satisfy a certain continuity condition; the lower bounds are obtained for the infimum of the above quantity over all choices of $\\mathbf{x}_k$ and $w_k$. Since the upper and lower bounds agree with respect to order, the complexity of quadrature in Besov spaces on the sphere is thereby established.
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Publication status
- Published
Journal
Journal of ComplexityISSN
0885-064XExternal DOI
Issue
4-6Volume
23Page range
528-552Pages
5.0Department affiliated with
- Mathematics Publications
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- No
Peer reviewed?
- Yes
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2012-02-06Usage metrics
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