Constructing a Gaussian kernel in the frequency domain
You can check numerically that fft(g) = G, that will be easier to analyze than the filtered result. At least you can easily plot it and identify if the coefficients are shifted, flipped, or scaled. One thing easy to do incorrectly is the determination of the frequency corresponding to each coefficient. I would strongly recommend that you first aproach the 1D version of your problem, here you have an example of 1D filtering I wrote for a related question.