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

We consider the numerical integration ${\rm INT}_d(f)=\int_{\mathbb{B}^{d}}f(x)w_μ(x)dx$ for the weighted Sobolev classes $BW^{r}_{p,μ}$ and the weighted Besov classes $BB_τ^r(L_{p,μ})$ in the randomized case setting, where $w_μ, \,μ\ge0,$ is the classical Jacobi weight on the ball $\Bbb B^d$, $1\le p\le \infty$, $r>(d+2μ)/p$, and $0<τ\le\infty$. 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 $p>1$.

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