Optimal randomized quadrature for weighted Sobolev and Besov classes with the Jacobi weight on the ball

• Published in: Journal of Complexity Vol. 73; p. 101691
• Main Authors: Li, Jiansong, Wang, Heping
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.12.2022
• Subjects: Deterministic case setting; Filtered hyperinterpolation; Optimal quadrature error; Randomized case setting; Weighted Besov classes; Weighted Sobolev classes; 41A55; 65D30; Optimal quadrature error; Weighted Besov classes; Randomized case setting; 65D32; Weighted Sobolev classes; Filtered hyperinterpolation; Deterministic case setting
• ISSN: 0885-064X; 1090-2708
• DOI: 10.1016/j.jco.2022.101691

We consider the numerical integrationINTd(f)=∫Bdf(x)wμ(x)dx for the weighted Sobolev classes BWp,μr and the weighted Besov classes BBτr(Lp,μ) in the randomized case setting, where wμ,μ≥0, is the classical Jacobi weight on the ball Bd, 1≤p≤∞, r>(d+2μ)/p, and 0<τ≤∞. For the above two classes, we obtain the orders of the optimal quadrature errors in the randomized case setting are n−r/d−1/2+(1/p−1/2)+. Compared to the orders n−r/d of the optimal quadrature errors in the deterministic case setting, randomness can effectively improve the order of convergence when p>1.