## Discretization of integrals on compact metric measure spaces.(English)Zbl 1471.46040

Summary: Let $$\mu$$ be a Borel probability measure on a compact path-connected metric space $$(X, \rho)$$ for which there exist constants $$c, \beta \geq 1$$ such that $$\mu(B) \geq c r^\beta$$ for every open ball $$B \subset X$$ of radius $$r > 0$$. For a class of Lipschitz functions $$\Phi : [0, \infty) \to \mathbb{R}$$ that are piecewise within a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric $$\rho$$ and the measure $$\mu$$ that for each positive integer $$N \geq 2$$, and each $$g \in L^\infty (X, d \mu)$$ with $$\| g \|_\infty = 1$$, there exist points $$y_1, \ldots, y_N \in X$$ and real numbers $$\lambda_1, \ldots, \lambda_N$$ such that for any $$x \in X$$, $\left| \int\limits_X \Phi (\rho (x, y)) g(y)\, \mathrm{d}\mu (y) - \sum\limits_{j = 1}^N \lambda_j \Phi (\rho (x, y_j)) \right| \leqslant CN^{-\frac{1}{2} - \frac{3}{2\beta}} \sqrt{\log N},$ where the constant $$C > 0$$ is independent of $$N$$ and $$g$$. In the case when $$X$$ is the unit sphere $$\mathbb{S}^d$$ of $$\mathbb{R}^{d + 1}$$ with the usual geodesic distance, we also prove that the constant $$C$$ here is independent of the dimension $$d$$. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound $$N^{-\frac{1}{2}} \sqrt{\log N}$$.

### MSC:

 46E36 Sobolev (and similar kinds of) spaces of functions on metric spaces; analysis on metric spaces 30L99 Analysis on metric spaces 54E45 Compact (locally compact) metric spaces

### Keywords:

cubature formulas; discretization; dimension-free estimates
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### References:

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