Linear Systems and Half-Line Integral Representations

We give an expression for the most general input-output map associated with the members of a certain important large family of causal time-invariant linear systems. This expression is a uniform limit of a convolution. We also give a necessary and sufficient condition under which the limit can be written as a convolution, and we relate this to the concepts of an impulse response and a q-response limit, and to the flawed concept of the Dirac impulse function. Key-Words: linear systems, impulse responses, time-invariant systems.