[bvw]

Well, that may be how you learned it! More simply, I think, the "convolution" part is just the multiplication of two polynomials. The z-transform is a physics and engineering thing, and fine for you to remember it, but the pertinent part is just the multiplication of polynomials.

A decimal number is itself a polynomial. For example 121 is really ( 1 times x0 ) plus ( 2 times x1 ) plus (1 times x2), where x0 is 10^0, x1 is 10^1 and x2 is 10^2.

And the factors of a polynomial, when the polynomial is power series, or follows some other positional rule -- those factors are just a sequence of numbers. A series of numbers in a set order.

What you call a convolution can also be called an "inner product".

[Lonesome in Massachussets]

My introduction to convolution was the classical continuous time version. These days everything is discrete time, so I just naturally think in those terms...

Honestly, when I first saw the problem, “What is the square root of 12345678987654321?”, I just about immediately recognized it as (de)convolution problem.

We tend to cast problems in familar terms.