= Autocorrelation function of a random field
{title2=$C(\mathbf s)$}
For a zero-mean <stationary Gaussian random field> $V$, the unnormalized <autocorrelation function of a random field> is
$$
C(h)=\mathbb E[V(x)V(x+h)].
$$
It depends only on the separation $h$, and $C(0)$ is the pointwise <variance>. If $C(0)>0$, the normalized correlation is $\rho(h)=C(h)/C(0)$; the two conventions have different units.
For a mean-square continuous field, its <spectral measure of a stationary random field> $\nu$ obeys $C(h)=\int e^{ik\cdot h}\nu(dk)$. If $C$ is <integrable>, an ordinary <power spectral density> exists and, in the angular-wavevector convention,
$$
S(k)=\int e^{-ik\cdot h}C(h)\,dh,\qquad
C(h)=\frac1{(2\pi)^d}\int e^{ik\cdot h}S(k)\,dk.
$$
Thus $\nu$ has density $S(k)/(2\pi)^d$. A spectral measure need not have a density: a spatially constant centered Gaussian field of <variance> $a>0$ has $C(h)=a$ and spectral measure $a\delta_0$, concentrated at zero <wavevector>. Calling every spectrum an ordinary <function> would miss this case.<Gaussian Processes for Machine Learning by Carl Edward Rasmussen and Christopher K. I. Williams>[Chapter 4, Section 4.1, Bochner representation and spectral-density distinction; Fourier factors explicitly chosen here]{ref}
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