Summary: We present sharp bounds on the minimal errors of linear estimators for multivariate integration and \(L_2\)- approximation. This is done for a random field whose covariance kernel is a tensor product of one-dimensional kernels that satisfy the Sacks-Ylvisaker regularity conditions [see J. Sacks and D. Ylvisaker, Ann. math. Stat. 39, 49-69 (1968; Zbl 0165.21505)].