If you plot your coefficients as if they were a discrete time signal, you ought to see multiplication appear as a blur I guess, but yeah IMO it's hard to interp

If you plot your coefficients as if they were a discrete time signal, you ought to see multiplication appear as a blur I guess, but yeah IMO it's hard to interpret for polynomials because polynomials are very abstract. I'd certainly look at the diagonal (or what I said is the "symmetric" way to write it) as the proof for "why" convolution appears in the sum of random variables, and I've never considered the blurring thing in that setting, but it actually makes a lot of intuitive sense for looking at "what" happens when you add distributions. Like just picturing repeated blurrings of a uniform distribution at the origin (i.e. start with a square and blur with a square over and over), it seems intuitively plausible that it will converge to something like a gaussian (i.e. that you'll get a bump of some kind), so that's neat.