Continuous LTI Systems Defined on $L^{p}$ Functions and ${cal D}_{L^{p}}^{\prime }$ Distributions:

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.