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arXiv · 2610.32699

Random nodes with positive weights: Richter-Tchakaloff rules and optimal positive quadrature

Sebastian Neumayer·Martin Schäfer·Tino Ullrich
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Abstract

We show that independent random points support exact quadrature rules with positive weights. For any $n$-dimensional space $V$ of integrable functions a constant multiple of $n\log n$ points sampled from a suitable density yields, with high probability, positive weights which integrate the functions in the space exactly. The weights stay close to the natural importance sampling weights according to the Lewis density with respect to $V$. They are the result of a weight adaptation procedure which requires an explicit $L_1$-norm discretization result that is proved on the way. As an application, we study numerical integration in reproducing kernel Hilbert spaces. Combining the random rules with a subsampling step, we show that positive quadrature rules with constantly oversampled nodes achieve the best possible worst-case error bound in terms of the singular numbers of the embedding into $L_2$.

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Sebastian Neumayer, Martin Schäfer, Tino Ullrich. 2026-10-23. Random nodes with positive weights: Richter-Tchakaloff rules and optimal positive quadrature. https://arxiv.org/abs/2610.32699

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