foutier transform question (math based)

Hi all,

I'm trying to work through a problem but am a bit stuck and I'd love to pick your (collective) brains for a moment. Thanks in advance for any suggestions/ideas.

I'm working on solving an expression of the form: result = IFFT(kernel*FFT(map)). kernel, map, and result are shown below in that order.

Now please direct your attention to map, which is the middle figure. In this case shown, the annulus is described by a constant nonzero value while the rest of the 2D array is zero. However, in reality each gridpoint in the annulus is described not by a scalar value but instead by a rank 2 tensor X_i,j

How might I approach solving the FFT of that "map"? Again, thanks in advance for any input!

[Image Analyst]

It's hard for me to tell with funky colors. I don't know what are high values and what are low values.