Optimal quadrature for weighted function spaces on multivariate domains
Jiansong Li, Heping Wang
TL;DR
The paper addresses numerical integration on the sphere $\bS^d$ and extensions to the ball and simplex for weighted Sobolev spaces with Dunkl weights and Besov spaces of generalized smoothness. It establishes sharp deterministic and randomized quadrature rates for $BW_{p,w}^r(\bS^d)$ and derives optimal upper bounds for $BB_\gamma^\Theta(L_{p,w}(\bS^d))$ via weighted least $\ell_p$ approximation and Monte Carlo, with rates depending on weight classes (e.g., $A_\infty$ or product weights). The analysis hinges on harmonic-analytic tools (Dunkl operators, spherical harmonics), weighted approximation theory (Jackson inequalities, Nikolskii-type inequalities), and Novak-style fooling function arguments for lower bounds. Extensions to the unit ball and simplex are obtained via weight-transfer maps, showing analogous optimal rates under comparable weight conditions. The results demonstrate that randomized algorithms can achieve faster convergence than deterministic ones when $p>1$, and they provide a comprehensive framework for optimal quadrature in weighted multivariate domains under broad weight assumptions, including doubling weights and $A_\infty$ weights.
Abstract
Consider the numerical integration $${\rm Int}_{\mathbb S^d,w}(f)=\int_{\mathbb S^d}f({\bf x})w({\bf x}){\rm d}σ({\bf x}) $$ 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 $\ell_p$ approximation and the standard Monte Carlo algorithm to obtain upper estimates of the quadrature errors which are optimal if $w$ is an $A_\infty$ 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 $\mathbb R^d$.