We consider Gaussian random fields on the product of a compact two-point homogeneous space cross the time, which are space isotropic and time stationary. We study regularity properties of these random fields in terms of function spaces whose elements have different smoothness in the space and time domain. Namely, we express the norm of the corresponding covariance kernel functions in terms of the summability of the associated spectral coefficients. Furthermore, we define an approximation method based on the truncation of the expansion related to the spectral representation of a given random field. The accuracy of this approximation is measured in the \(L^p\) sense. Finally, we model a space–time dataset of ozone concentrations in Mexico City using a seasonal temporal covariance function constructed through an expansion of Jacobi polynomials. We find that we need relatively few Jacobi polynomials to get the best fit to the data in terms of the deviance information criterion. We discuss the characteristics of this model, including seasonality, decay and approximate conditional independencies.

我们考虑一个紧凑的两点齐次空间的乘积跨时间的高斯随机场，该空间是各向同性的且时间是平稳的。我们根据函数空间研究这些随机字段的规则性，这些函数空间的元素在时域和空间上具有不同的平滑度。即，我们根据相关频谱系数的可和性来表达相应协方差核函数的范数。此外，我们基于与给定随机场的频谱表示有关的扩展的截断定义了一种近似方法。该近似值的精度以\（L ^ p \）表示感。最后，我们使用通过扩展Jacobi多项式构造的季节时间协方差函数对墨西哥城的臭氧浓度时空数据集进行建模。我们发现，我们需要相对较少的Jacobi多项式，以根据偏差信息准则对数据进行最佳拟合。我们讨论了该模型的特征，包括季节性，衰减和近似条件独立性。