We might see something similar to biological evolution happen here. The most complicated structures repurposed components that evolved for other reasons. Bird flight wouldn’t have happened without feathers, but feathers could not have evolved for the sake of flight–it’s just too big a step. Instead, they had already evolved for something else, like insulation, and thus were already available for use with flight.
So perhaps something similar could take place here–train a net on something that more directly learns an FFT, and then when it is trained on image data later, it has the FFT available for use already. If the FFT is useful for that purpose, the image training will refine it further, possibly opening it up for yet more applications.
Put another way, all AI training is just some form of gradient descent. But for some training data, there is just no path from the current position to some deeper minima far out. Other training data may however unlock a smooth path from here to there. Once there, the data that was previously stuck now has more options available.