I read a paper, and am confused about the following:
Suppose $W$ is an operator with impulse response (IR) $w$. And suppose $w^n$ is the IR of $W^n$.
My question is the following:
It is $w^n(T)*w(T)$ from the right hand side. How could I show it is equal to the left hand side?
If systems $A$ and $B$ have impulse responses $a(t)$ and $b(t)$, then the cascaded system $AB$ has impulse response $a(t) * b(t)$. This is a pretty basic fact in linear system theory.
The system $W^{n+1} = WW^n$ is simply the system $W^n$ cascaded with the system for $W$.
So the impulse response for the cascaded system $W^{n+1} = WW^n$ is simply the convolution of the impulse response of $W^n$ with the impulse response of $W$, i.e. $w^{(n+1)}(T) = w(T) * w^{(n)}(T)$.