Optimal quadrature for weighted function spaces on multivariate domains

2024/12/23 by Jiansong Li, Heping Wang, Li, Jiansong +1
Mathematics · #Mathematical Approximation and Integration #Approximation Theory and Sequence Spaces #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2412.17546

Abstract

Consider the numerical integration \rm Int_{\mathbb S^d,w}(f)=∫_{\mathbb S^d}f(\bf x)w(\bf x)\rm dσ(\bf x) for weighted Sobolev classes BW_p,w^r(\mathbb S^d) with a Dunkl weight w and weighted Besov classes BB_γ^Θ(L_p,w(\mathbb S^d)) with the generalized smoothness index Θ and a doubling weight w on the unit sphere \mathbb S^d of the Euclidean space \mathbb R^d+1 in the deterministic and randomized case settings. For BW_p,w^r(\mathbb S^d) we obtain the optimal quadrature errors in both settings. For BB_γ^Θ(L_p,w(\mathbb S^d)) we use the weighted least ℓ_p approximation and the standard Monte Carlo algorithm to obtain upper estimates of the quadrature errors which are optimal if w is an A_∞ weight in the deterministic case setting or if w is a product weight in the randomized case setting. Our results show that randomized algorithms can provide a faster convergence rate than that of the deterministic ones when p>1. Similar results are also established on the unit ball and the standard simplex of \mathbb R^d.

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