Does fading memory mean impulse response with finite support? Given a continuous-time impulse response $h(t)$, bandlimited to $B$.
The discrete-time $h[n]=h(n/(2B))$ is then a unique and perfect representation of $h(t)$ and a discrete-time system $h[n]$ is then a perfect representation of $h(t)$ (impulse variance).
However, $h[n]$ generally requires infinitely many terms. I am looking for a name of systems $h(t)$ where $h[n]$ has finite terms (that means $h(t)$ has finite support) or can arbitrarily well approximated by finite terms.
Obviously the name for $h[n]$ (the discrete-time equivalent) is a FIR filter.
Is a system with fading memory the right term?
If not, why not and what is the right term?
PS: FIR seems to be a terms almost exclusively used for discrete-time systems. Why does it not exist for for continuous-time systems?
What you want is theoretically impossible: if $h(t) \neq 0$ only on a finite interval, then it is not bandlimited. Fortunately, all real-world signals and systems have the property that $H(f) \rightarrow 0$ as $|f| \rightarrow \infty$. This allows you to approximate $h(t)$ very closely with only a finite number of samples. For engineering purposes, this is often more than enough.
Note that the property above means that $h(t)$ has finite energy and is absolutely integrable (it is in $L_1$). For the math to work out, it is often also required that $h(t)$ is in $L_2$, or that its square is absolutely integrable. These are the terms that are commonly used to describe these kinds of signals and systems.