Partially translation invariant linear systems

The subject of this paper is the analysis of abstract linear systems in several variables, i.e. continuous linear mappings from Banach space valued test functions as inputs to Banach space valued distributions as outputs, N being partially translation invariant in the sense that it commute with translations from a given closed subgroup G of the additive group. The classical convolution representation of the invariant case generalizes to a Fourier series representation whose coefficients N_k are linear systems commuting with translations from the vector subgroup lin G generated by G. Thus N is essentially a "Fourier superposition" of classical systems. It is shown that causality of N with respect to a subsemigroup P of implies causality of the N_k which in turn corresponds to carrier conditions on the N_k.Dedicated to Professor Heinz König on the occasion of his 50th birthday.