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

Abstract

We consider the numerical integrationfor the weighted Sobolev classes BW r p,µ and the weighted Besov classes BB r τ (Lp,µ) in the randomized case setting, where wµ, µ ≥ 0, is the classical Jacobi weight on the ball B d , 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.