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Any explanation should start by saying a convolution is just a weighted average.

Well it is, sort of, but treating one of the functions as weights and the other as the thing that is averaged misses out on the symmetry f*g = g*f

Averages are normalized. Convolutions only are if they integrate to 1.

What are you using non normalized weighted averages for?

I'm not! Averages should normalize the weights. But there are plenty of reasons to define convolutions where neither signal is normalized.

What are they?