Notes on representation theorems for linear discrete-space systems

The cornerstone of the theory of discrete-space single-input single-output linear systems is the idea that every such system has an input-output map H that can be represented by a convolution or the familiar generalization of a convolution. This thinking involves an oversight which, for the case of bounded inputs mapped continuously into bounded outputs, was recently corrected by adding an additional term to the representation. Here we give a more general result that addresses an important larger family of inputs