= Spectral measure of a stationary random field
{title2=$C(\zeta)=\int_{\mathbb R}e^{i\nu\zeta}F(d\nu)$}
= Spectral measures of a stationary random field
{synonym}
A second-order stationary complex random field on the line has uncentered correlation $C(\zeta)=\mathbb E[E(z+\zeta)\overline{E(z)}]$. When this correlation is continuous, its <Fourier transform> is a finite positive measure $F$ characterized by $C(\zeta)=\int e^{i\nu\zeta}F(d\nu)$. Its total mass is $\mathbb E|E(z)|^2$. If it has a density, $S(\nu)=(2\pi)^{-1}\int C(\zeta)e^{-i\nu\zeta}d\zeta$. A constant mean $m$ contributes $|m|^2\delta_0$; the remaining measure is the covariance spectrum. This uncentered convention is useful for <coherent and diffuse wave fields>.
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