A note on impulse response for continuous, linear, time-invariant, continuous-time systems

In the paper by Sandberg in 2003, he proved that, even if a continuous, linear, time-invariant, continuous-time system admits an impulse response, such a response does not always give a complete description of the system. In this paper, a theorem by Schwartz is used to define an impulse response under almost general assumptions, and to understand what we really know about two systems with the same impulse response. These results are applied to a survey of systems (significant by themselves and as leading examples), showing that, apart from three classes of exceptions, all of them are completely described by their impulse response. Concerning the first two classes of exceptions, counterexamples were given by Sandberg; concerning the remaining third class, a counterexample is deduced here from Sandberg's results.