Question

Convolution: whether it is convolution of functions, measures or sequences, it is often defined by giving an explicit formula for the resulting function (or measure, etc.). While this definition makes calculations with convolutions relatively easy, it gives little intuition into what convolution really is and often seems largely unmotivated. In my opinion, the right way to define convolution (say, of two finite complex Radon measures on an LCA group $G$, which is a relatively general case) is as the unique bilinear, weak-* continuous extension of the group product to $M(G)$ (the space of measures as above), where $G$ is naturally identified with point masses. Then one can restrict the definition to $L^1 (G)$ and get the well known explicit formula for convolution of functions. Of course, a probabilist will probably prefer to think of convolution as the probability density function associated to the sum of two independent absolutely continuous random variables. And there are other possible alternative definitions (see this Mathoverflow discussion). But the formula definition is really the hardest one to get intuition for, in my opinion.