On the Representation of Shift-Varying Linear System Maps

Abstract

We give an expression for the most general continuous linear input-output map that takes the space of (Lebesgue) integrable functions into itself. This expression is a function-space limit of an integral. As an application, a representation result is given for an important family of linear maps that take the space of bounded measurable functions into itself and satisfy a certain continuity condition, and it is noted that no such result holds for a general continuous linear maps that takes the space into itself and meets the continuity condition. Some related material concerning engineering education is also given.