Time Invariance

Abstract

Unlike the concepts of causality, stability, state, etc. which are well defined in an arbitrary resolution space, time-invariance and the various representations for such operators are well defined only in a restricted class of resolution spaces termed uniform resolution spaces. The purpose of this chapter is the formulation of the uniform resolution space concept and the study of time-invariant operators defined on such spaces. For the most part, an assumption of time-invariance yields little in strengthening the system theoretic results we have already obtained, but it allows a number of powerful operator representation theorems to be invoked for the simplified description of the various properties we have already studied. As such, the chapter is devoted primarily to the formulation of appropriate Fourier and Laplace transformations for time-invariant operators with their explicit interpretation in a systems context left to the problems of the last section.