Continuous LTI Systems Defined on $L^{p}$ Functions and ${cal D}_{L^{p}}^{prime}$ Distributions: Analysis by Impulse Response and Convolution

摘要

?In this paper, it is shown that every continuous linear time-invariant system ${cal L}$ defined either on $L^{p}$ or on ${cal D}_{L^{p}}^{prime}$ $(1leqslant pleqslantinfty)$ admits an impulse response $Deltain{cal D}_{L^{p^{prime}}}^{prime}$ ($1leqslant p^{prime}leqslantinfty$ , $1/p+1/p^{prime}=1$ ). Schwartz' extension to ${cal D}_{L^{p}}^{prime}$ distributions of the usual notion of convolution product for $L^{p}$ functions is used to prove that (apart from some restrictions for $p=infty$), for every $f$ either in $L^{p}$ or in ${cal D}_{L^{p}}^{prime}$ , we have ${cal L}(f)=Deltaast f$. Perspectives of applications to linear differential equations are shown by one example.