Adjusted Euler–MacLaurin predictor for integrating smooth spatial processes

We consider the problem of predicting integrals of a spatial stationary process $Z$ over a unit square. We construct predictors based on a systematic sampling of size $m^2$ by approximating the existing mean squared derivatives of the process in the two-dimensional Euler–MacLaurin formula by finite differences up to some appropriate order. We show that if the spectral density satisfies $f_{Z}(\omega) =o(|\omega|^{-p})$ for any fixed positive integer $p$, the corresponding mean squared error is of order $m^{-p}$.