零均值高斯测度的正交性：充分稠密采样
On the orthogonality of zero-mean Gaussian measures: Sufficiently dense sampling
AI总结：
本文研究平稳随机函数在子集上采样时两个零均值高斯测度的等价与正交性，给出各向同性条件下的等价判据，并证明当采样点到原点距离稠密时正交性可恢复。
摘要：
对于在$\mathbb{R}^{d}$的子集$D$上采样的平稳随机函数$\xi$，我们考察与$\xi$相关的两个零均值高斯测度$\mathbb{P}_{1}$和$\mathbb{P}_{2}$的等价性与正交性。我们给出了各向同性的类比结果，即$\mathbb{P}_{1}$和$\mathbb{P}_{2}$的等价性与从$D$到$\mathbb{R}^{d}$的$\mathbb{P}_{1}$和$\mathbb{P}_{2}$的协方差函数之差的平方可积扩展的存在性相关联。我们证明，当从$D$中的点到原点的距离集合在非负实数集合中稠密时，$\mathbb{P}_{1}$和$\mathbb{P}_{2}$的正交性可以被恢复。
英文摘要：
For a stationary random function $ξ$, sampled on a subset $D$ of $\mathbb{R}^{d}$, we examine the equivalence and orthogonality of two zero-mean Gaussian measures $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ associated with $ξ$. We give the isotropic analog to the result that the equivalence of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ is linked with the existence of a square-integrable extension of the difference between the covariance functions of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ from $D$ to $\mathbb{R}^{d}$. We show that the orthogonality of $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ can be recovered when the set of distances from points of $D$ to the origin is dense in the set of non-negative real numbers.
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