= Gaussian shell covariance
{c}
{title2=$G_>(x-y)$}
For a centered shell <Gaussian measure> of a real <scalar field> with positive shell kernel $K(p)=\kappa p^2+r$, its <Gaussian functional integral> gives $\langle\widetilde\phi(p)\widetilde\phi(q)\rangle=(2\pi)^D\delta^{(D)}(p+q)/K(p)$ on the shell. Inverse <Fourier transforms> therefore yield $G_>(x-y)=\int_{\Lambda/b<|p|\leq\Lambda}d^Dp\,(2\pi)^{-D}e^{ip\cdot(x-y)}/K(p)$. The kernel must be positive throughout the shell. <Translation invariance> makes the separation essential; a numerator involving $x$ alone is valid only with $y=0$ or if $x$ is redefined as the separation.
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