It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of the samples of a Gaussian process $X$ is controlled by a certain nuclear dominance condition. However, it is less clear how to identify a "small" set of functions (not necessarily a vector space) that contains the samples. This article presents a general approach for identifying such sets. We use scaled RKHSs, which can be viewed as a generalisation of Hilbert scales, to define the sample support set as the largest set which is contained in every element of full measure under the law of $X$ in the $\sigma$-algebra induced by the collection of scaled RKHS. This potentially non-measurable set is then shown to consist of those functions that can be expanded in terms of an orthonormal basis of the RKHS of the covariance kernel of $X$ and have their squared basis coefficients bounded away from zero and infinity, a result suggested by the Karhunen-Lo\`{e}ve theorem.

翻译：已知某个复制核心Hilbert空间(RKHS)的标本由特定核支配地位条件控制,但不清楚如何确定包含样品的“小型”功能组(不一定是矢量空间),本文章为确定这类组别提供了一个一般方法。我们使用可视为对Hilbert尺度的概括的按比例规模的RKHS,将样品支持组确定为最大数据集,根据法律,在按比例收集的RKHS所引出的全部计量值$x$的每个要素中,该数据集包含在最大要素中。这一潜在不可计量的数据集由这些功能组成,这些功能可以以0.X美元共振核心的RKHS的正态基础为基础加以扩展,并将它们的正方基系数从零值和宽度上绑开,这是Karhunen-Lo ⁇ eev theorem建议的结果。