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\begin{equation}

\begin{split}

COV(x,k)&= E[x k]-E[x]E[k]

\end{split}

\end{equation}

If ##x## and ##k## are Foureir conjugates and ##f(x)## and ##\hat{f}(k)## are Gaussian distributions, how does that affect the covariance?

This is not a homework problem. I am just trying to understand the covariance of Fourier conjugates, particularly for Gaussians.