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An important question in the theory of approximate integration is to study the conditions on the nodes $x_{k,n}$ and weights $w_{k,n}$ that allow an estimate of the form $$ \sup_{f\in \mathcal{B}_\gamma}|\sum_k w_{k,n}f(x_{k,n})-\int_\mathbb{X} fd\mu^*| \le cn^{-\gamma}, \qquad n=1,2,\cdots, $$ where $\mathbb{X}$ is often a manifold with its volume measure $\mu^*$, and $\mathcal{B}_\gamma$ is the unit ball of a suitably defined smoothness class, parametrized by $\gamma$.
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