Periodically Invariant Linear Systems

Abstract

The topic of this paper is the analysis of abstract linear systems on a locally compact group G, that is, of continuous linear operators $N:\mathfrak{D}(G) \to \mathfrak{D}'(G)$. Here, $\mathfrak{D}(G)$ denotes the space of test functions on G with its inductive limit topology as introduced by Bruhat, Maurin and Kac and $\mathfrak{D}'(G)$ the space of distributions on G with the weak topology. We call such a linear system N periodically invariant with respect to a given closed subgroup $\Gamma $, if N commutes with translations from $\Gamma $. Periodically invariant systems are of interest, e.g., in the theory of electrical networks with periodically varying parameters or in process control theory. Under the assumption that the quotient group ${G / \Gamma }$ is compact, a Fourier series representation for $\Gamma $-periodic distributions on G is derived. From this we conclude via the Schwartz kernel theorem that the classical convolution representation for a translation invariant system $(\Gamma = G)$ generalizes to a Fourier superposition $\Sigma _\chi N_\chi $ of translation invariant systems $N_\chi $, where summation runs over all characters $\chi \in \hat G$ vanishing on $\Gamma $. Finally, it is proved that N is causal with respect to a semigroup $P \subset G$ if and only if all individual systems $N_\chi $ are causal.