Continuous LTI systems defined on Lpfunctions and D'Lp distributions: Analysis by impulse response and convolution

Abstract

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