Continuous LTI systems defined on Lp functions and D'L p 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
'
_L

p
(1les
p
lesinfin) admits an impulse response Deltaisin
D
'
_L

p
' (1les
p
'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.