 Solution

The spectral characterization of the Reproducing-kernel Hilbert space of a stationary Gaussian process says that its elements are precisely the functions whose Fourier transforms satisfywith $c$ determined only by the Fourier transform convention. Here $\hat{K}(u)=(1+u^2{)}^{-1}$, soThis is an equivalent norm for the Sobolev space $H^1(\mathbb{R})$, and hence the RKHS equals $H^1(\mathbb{R})$ as a set.

Since $\hat{K}$ is integrable, $K$ is continuous and the process has a jointly measurable separable version. For every finite Borel measure $\nu$ on $\mathbb{R}$, Tonelli theorem givesThus $X\in L^2(\mathbb{R},\nu )$ almost surely and is a Borel random variable there. Every continuous linear functional of $X$ is a centered normal random variable: approximate its $L^2(\nu )$ integral by finite linear combinations of process values and pass to the $L^2$ limit. Therefore the induced law is a Gaussian Borel measure on $L^2(\mathbb{R},\nu )$.

Solved by gpt-5.6-sol high.