Convolution systems

Since the introduction of the behavioral approach to systems theory by Willems [13, 14, 15, 16, 17] the main emphasis has been on discrete time systems. Papers dealing with continuous time systems generally have assumed from the start that a set of differential equations was given. In his thesis Soethoudt has given necessary and sufficient conditions on the behavior of a continuous time system to be the solution set of a system of ordinary, linear differential equations with constant coefficients (Soethoudt [12]). In this paper I choose a different starting point. Given a behavior, linear and time invariant, is there something that one can derive about the equations that elements in the behavior have to satisfy, what properties can be derived from these assumptions alone, are that other requirements that have to be imposed on the behavior to be able to derive meaningful results? In an earlier with Soethoudt [8] we showed that some restrictions have to be made on the function class. Further to be able to describe a behavior by a set of continuous equations, it should be closed in a reasonable topology. In this paper I restrict myself to locally square integrable functions on the real axis (in a joint paper with Soethoudt [9] we treated some questions for functions on a right half axis). The paper is organized as follows: in § 1 I show that a behavior can be described as the solution set of a a priori infinite family of convolution equations. § 2 the easiest case is treated: behaviors consisting of real valued functions. This section is based on the theory of mean periodic functions, as introduced by Delsarte [2] and developped by Schwartz [11], Kahane [5], Ehrenpreis [3] and others. If possible I have gathered the more technical details, not essential for the development of the ideas in the paper, in appendices.