Expectation value of Fourier conjugates

[redtree]

I understand that the Uncertainty Principle relates the variances of Fourier conjugates. I am having trouble finding: 1) the mathematical relationship between the expectation values of Fourier conjugates generally; 2) and then specifically for a normalized Gaussian. Any suggestions or insights?

 

[BvU]

Work your way through a textbook on Fourier transforms
1) is there any ?
2) FT of Gaussian is Gaussian. Product of their $\sigma$ is constant. expectation values are not related but I like to be shown wrong