discrete FT vs continuous FT question

Hi all,

I have a two part question concerning the fftn (dFT) algorithm on Matlab.

The problem I am working through is as follows. I have an image in real space. I perform FT on the image. Next I multiply it with a kernel in fourier space. And then I inverse fourier transform that product back into real space.

The analytic expression for the result is known. These equations are derived from working through a cFT. The result I get from Matlab fft2/fftn is "close" but not the same as the analytic expression. I've attached an image showing this discrepancy below.

Next I tried something different. Instead of using a image that has angular variation, I used used an image that was constant: ones inside the annulus and zeros outside. The result produced by the fft2/fffn method matched very accurately that predicted by analytic equations. *By the way the image of the result produced by the fourier method is the real component.

Initially, I had thought that the dFT did not deal well with sharp boundaries. But the second experiments shows otherwise. I can't seem to wrap my head around what exactly is going wrong in the first experiment. I suspect it is the angular variation? But it doesn't make sense to me. I know this question is really open ended but I thought perhaps someone has encountered something like this before. I'd really appreciate any insight, comments, or ideas.