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 \(N^{-\frac{1}{2}} \sqrt{\log N}\).


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
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