Optimal order integration on the sphere

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This paper reviews some recent developments in cubature over the sphere $S^2$ for functions in Sobolev spaces. More precisely, for an $m$-point cubature rule $Q_m$ we consider the worst-case (cubature) error, denoted by $E(Q_m;H^s)$, of functions in the unit ball of the Sobolev space $H^s=H^s(S^2)$, with $s>1$. The following recent results are reviewed in this paper: For any sequence $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of positive weight $m(n)$-point cubature rules $Q_{m(n)}$, where $Q_{m(n)}$ integrates all spherical polynomials of degree $\\leq n$ exactly, the worst-case error in $H^s$ satisfies the estimate $E(Q_{m(n)};H^s)\\leq c_s n^{-s}$ with a universal constant $c_s>0$. Whenever $m(n)=O(n^2)$ we deduce $E(Q_{m(n)};H^s)\\leq c_s m(n)^{-s/2}$, where the constant $c_s$ now depends on the constant in $m(n)=O(n^2)$. This rate of convergence is optimal since it has also been shown that there exists a universal constant $\\tilde{c}_s>0$ such that for any $m$-point cubature rule $Q_m$, the worst-case error in $H^s$ with $s>1$ satisfies $E(Q_m;H^s)\\geq\\tilde{c}_s m^{-s/2}$. For example, sequences $(Q_{m(n)})$ of positive weight product rules with $m(n)=O(n^2)$ achieve the optimal order of convergence $O(m(n)^{-s/2})$. So too, if the weights are all positive, do sequences $(Q_{m(n)})$ of interpolatory cubature rules based on extremal fundamental systems.