Optimal cubature formulas on compact homogeneous manifolds
Let \(\mathbb{M}^d\) be a~two-point compact homogeneous manifold in \(\mathbb{R}^{d+1}\) and let \(W_{\infty}^r(\mathbb{M}^d)\) stands for the class of \(r\)-times differentiable functions \(f:\mathbb{M}^d\to\mathbb{R}\) with \(\|f\|_{\infty}\leq 1\). The main result states that for any \(r>0\) and \(\varepsilon>0\) \[ \inf_{\{x_1,\dots,x_n\}\subset\mathbb{M}^d}\inf_{\{\alpha_1,\dots,\alpha_n\}\subset\mathbb{R}}\;\sup_{f\in W_{\infty}^r(\mathbb{M}^d)}\left|\int_{\mathbb{M}^d}fd\nu-\sum_{i=1}^n\alpha_kf(x_k)\right|\geq Cn^{-r/d}(\log n)^{-(1+\varepsilon)}, \] where \(C\) is a constant depending only on~\(\varepsilon\).
scientific article; zbMATH DE number 5602668
scientific article