Continuous LTI Systems Defined on Lp Functions and DLpprime Distributions: Analysis by Impulse Response and Convolution.

In this paper, it is shown that every continuous linear time-invariant system L defined either on L(p) or on D'(Lp) (1 <= p <= infinity) admits an impulse response Delta is an element of D'(Lp') (1 <= P' <= infinity, 1/p + 1/p' = 1). Schwartz' extension to D'(Lp) distributions of the usual notion of convolution product for LP functions is used to prove that (apart from some restrictions for p = infinity), for every f either in L(p) or in D'(Lp), we have L(f) = Delta * f. Perspectives of applications to linear differential equations are shown by one example.