= Spectral tensor
{title2=$\widehat C_{ij}(\mathbf k)$}
For a statistically homogeneous vector <random field>, its spectral tensor is the <Fourier transform> of the componentwise two-point <correlation> $C_{ij}(\mathbf r)=\mathbb E[B_i(\mathbf x+\mathbf r)B_j(\mathbf x)]$. It distributes cross-correlations between components over <wavevectors>. With $\widehat B(\mathbf k)=\int B(\mathbf r)e^{-i\mathbf k\cdot\mathbf r}d^dr$, homogeneity gives $\mathbb E[\widehat B_i(\mathbf k)\widehat B_j(\mathbf k')]=(2\pi)^d\delta(\mathbf k+\mathbf k')\widehat C_{ij}(\mathbf k)$, understood as an identity of <distributions> for an infinite stationary field.
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