Error Propagation when Convolving Uncorrelated or Correlated Data

It may be necessary to convolve data with some kernel, which is often Gaussian. For example the resolution of a spectrum or map has to smoothed to increase the signal to noise or match the resolution of another data set. What to do if the dataset comes with uncertainties for each value. Here I derive rigorously how to propagate these uncertainties for any kernel and for Gaussian kernels. For uncorrelated data it is fairly straight forward, but for correlated data it is more involved. The result presented here lets you estimate the uncertainties of n-dimensional data that has been convolved with a Gaussian even when the data is correlated data with Gaussian covariances. In short, the result is that you need to convolve the uncertainties with the square of your kernel. If the data is correlated there is a scale factor that has to be applied to the uncertainties compared to the uncorrelated case assuming Gaussians for the correlation and convolution.