= Reproducing-kernel Hilbert space of a stationary Gaussian process
{c}
A stationary <Gaussian process> has a covariance <positive-semidefinite kernel> $k(x,y)=K(x-y)$. In the integrable <Fourier transform> setting on $\mathbb R^d$, use the convention
$$
K(h)=\frac1{(2\pi)^d}\int e^{iu\cdot h}\widehat K(u)\,du,
$$
with nonnegative integrable <spectral density> $\widehat K$. Its <Reproducing kernel Hilbert space> has the weighted <Fourier transform> norm
$$
\|f\|_{\mathcal H_K}^2=\frac1{(2\pi)^d}\int_{\{\widehat K>0\}}\frac{|\widehat f(u)|^2}{\widehat K(u)}\,du.
$$
The admissible <functions> have finite displayed norm and $\widehat f=0$ <almost everywhere> where $\widehat K=0$. That support condition is essential: a zero spectral weight cannot supply a missing frequency by treating an undefined quotient as zero. Small positive spectral weights instead make corresponding components expensive.
The norm describes deterministic <functions> associated with the covariance <positive-semidefinite kernel>. It does not assert that a random sample path of the <Gaussian process> has finite norm in this space. Regularization and random-path variation therefore describe different aspects of the same covariance model.<Gaussian Processes for Machine Learning by Carl Edward Rasmussen and Christopher K. I. Williams>[§6.1, equation 6.4; §6.2, equations 6.12–6.14]{ref}
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