the convolution function concept is the same as multiply function but in different domains. in fact the convolution concept is mixed with both frequency and time domain concepts. I think if you want to know where does convolution came from, you should find a conceptual relation between frequency and time domain first and then you get information about the base of fourier and laplas theory.
rgds.
Convolution is not a function, but an OPERATION, like ADDITION, SUBTRACTION, ....
In physics, a problem with distributed source is usually hard to solve, while a problem with point source is easy. So, a standard way to solve the complicated problem is to solve the problem with a point source first and then convolve the solution with the distributed source to obtain the solution of the original problem.
My question is about the author of Convolution operation. How is he desenvolved this operation? What was his origin point? ( Like Laplace has discovered the S transfor preparing a class for his student)