# Quadrature in Besov spaces on the Euclidean sphere

Hesse, K, Mhaskar, H N and Sloan, I H (2007) Quadrature in Besov spaces on the Euclidean sphere. Journal of Complexity, 23 (4-6). pp. 528-552. ISSN 0885-064X

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## Abstract

Let $q\\geq 1$ 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.

Item Type: Article School of Mathematical and Physical Sciences > Mathematics Kerstin Hesse 06 Feb 2012 21:18 11 Apr 2012 09:51 http://sro.sussex.ac.uk/id/eprint/30722