Optimal lower bounds for cubature error on the sphere S2

We show that the worst-case cubature error $E(Q_m;H^s)$ of an $m$-point cubature rule $Q_m$ for functions in the unit ball of the Sobolev space $H^s=H^s(S^2)$, $s>1$, has the lower bound $E(Q_m;H^s) \\geq c_s m^{-s/2}$, where the constant $c_s$ is independent of $Q_m$ and $m$. This lower bound result is optimal, since we have established in previous work that there exist sequences $(Q_{m(n)})_{n\\in\\mathbb{N}}$ of cubature rules for which $E(Q_{m(n)};H^s) \\leq \\tilde{c}_s (m(n))^{s/2}$ with a constant $\\tilde{c}_s$ independent of $n$. The method of proof is constructive: given the cubature rule $Q_m$, we construct explicitly a bad function $f_m\\in H^s$, which is a function for which $Q_m f_m=0$ and $\\|f_m\\|_{H^s}^{-1} | \\int_{S^2} f_m(\\mathbf{x}) d\\omega(\\mathbf{x})| \\geq c_s m^{-s/2}$. The construction uses results about packings of spherical caps on the sphere.