This work considers Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function (Stein, 1999, Section 6.7) with Fourier coefficients $\phi$($\alpha$^2 + j^2)^(--$\nu$--1/2). Convergence rates are studied for the joint maximum likelihood estimation of $\nu$ and $\phi$ when the data is sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a ''deterministic'' element of a continuous Sobolev space is also considered, suggesting that bounding assumptions on some parameters can lead to different estimates.

翻译：这项工作考虑了Gaussian过程的内插,其间有一段时间化版的 Mat_'e}rn 共变函数(1999年, Stein, 1999年, 第6.7节),以Fourier 系数 $\phi$($\pha$2 + j ⁇ 2) }(-$\nu$--1/2)。当数据按照模型抽样时,对一致率进行了联合最大可能性估计,即$\nu$和$/phi$。还用固定和估计参数分析了平均集成的误差,表明最大概率估计与已知地面真相的误差相同。最后,也考虑了所观察到的函数是连续的Sobolev空间的“非确定性”元素的情况,表明某些参数的捆绑假设可能导致不同的估计。