(sum_i ai x^i)(sum_j bj x^j) when you expand that, the coefficient of x^k is the sum of aibj such that i+j=k which is discrete convolution.
Set x=10 and you're dealing with decimal representation of numbers.
Set x=10 and you're dealing with decimal representation of numbers.
...and you no longer have a function, only the image of a function at a specific point of its domain.
Convolutions are functions, not functions evaluated at a specific, and cherry-picked point.