If I have a rect function , and I convolute it with it's self, I get a triangle function. If I convolute with a rect function again, I get a bell-curve. I can continue, so long as I know how to convolute two functions.
When I learnt this a few years back on my Physics course, I began to wonder if this was possible, or had applications in probability, where the rect function can act as a function for uniform deviates. Specifically I thought of dice rolls, which can be expressed as a uniform deviate convoluted with a comb of Dirac function. If I wanted to know the distribution of say 3 six sided dice (or some arbitrarily complicated roll of dice), I could just multiply the appropriate Fourier transform, then transform it back.
The way I was taught about Fourier transforms (by my physics lecturer) though was that Frequency maps to Time and Distance maps to Angles when you transform.
What does the Fourier transform of probability map to? How can it help my endeavour to work out dice probability distributions?