ROCZNIKI POLSKIEGO TOW ARZYSTW A MATE MAT Y CZN EGO Séria I: PRACE MATEMATYCZNE X V (1971)
A N N A L ES SOCIETATIS MATHEMATICAL POLONAE Series I: COMMENTATIONES MATHEMATICAE X V (1971)
J. S
zelm e c z k a(Lublin)
On Laplace transformation in spaces D '
l i1. The purpose of this note is to investigate the Laplace transform of 31-integrable distribution of one variable, following the idea of [2], where integral transforms with respect to a real-valued kernel are con
sidered. Adopting-the notation of [3] we define the integral of T eB ' as the value
CO
( / Ttdt) M = / Tt [<p{t—u)]du,
— O O
if the integral at the right-hand side of this equation exists for all (pel) (see [2]). Obviously, j T tdt is a constant distribution.
Now, let us suppose the carrier of T e B ’ is contained in <0, 00 ) and the integral (LT)S — je~stTtdt exists for complex s with E e s > cr0. Then it is a constant distribution for all s with Ее s > a0, i.e.
00
(LT)s{(p) = ( J e~stTtdt ) ( 9
9) = ( LT)(s
) J<p(u)du.
— O O
The function (LT){s) will be called the Laplace transform of T. It is easily seen that {LT)S is given by any of the two formulae:
OO
( L T ) M
=
J— OO
and
t —a
{LT)s{cp) = lim Tt \e~st f <p{u)dv\*
a -* -o o t _ b
o —>00
For example, if T is defined by a locally integrable function pos-
OO
sessing the Laplace transform, i.e. T(<p) = J z(t)(p(t)dt, then (LT)(s)
oo 0
= fe~stz(t)dt.
0
58 J. S z e l m e c z k a
If T = ôt(j is the Dirac distribution at t0, i.e. T(<p) = <p(t0), we get (LT) (s) = e~st0.
Let M{u) be an even convex function, M(u) = 0 if and only if
M (и) M (u)
и = 0, lim ---= oo, l i m --- = 0 and let N (u) be complementary
w-*-oo 'll u—>0 W'
to M(u) in the sense of Young. Moreover, let both M(u) and N(u) satisfy condition (A2) for all и and let
OO
f N{e~u) d u < oo.
— OO
We denote by D'M the space of Ж-integrable distributions, consisting П
of all distributions of the form T = ffz\k\ where z(k) is the fc-th derivative
*= i
of a function zk belonging to the Orlicz space L M(0 , oo) (see [2]).
П
2. Let T = £ z j f \ where zkeLM( 0 , oo). Then e~^z{t)^D'L\ and (LT)(s)
n oo fc= 1
= ffiskfe~'9tzk(t)dt for B es > 0 .
f t = 1 0
P roof. We may limit ourselves to T = z(k), where zeL M{0, oo).
We have any (pel) ,
к /7 \ oo
/ \e stzW[(p{t—ii) ]\d u ^ c ( « Г/ e ®es|s(^)| \dt.
— oo v = 0 ' ' 0
where c > 0 is a constant dependig on (p only.
But by the Young inequality,
f e~tResj z ( t ) l d t ^ f M[z(t)]dt + j N [ e - mes]dt.< oo.
0 0 0
Hence e~stz{t)eD'Li for B e s > 0 . Moreover,
{
l t)
s(<
p) = ( - i ) ‘ у ( Ь (-«)•*(*){ / v '* - ’(<-«)<*»} at
V=0 ' ! — oo
CO oo
= J e~s*zOf)d£-J q>{u)du
о о
for Be s > 0 . П
3. Let T = where z%LM{0, oo) let eGt 00
(L~l(p)(t) = — f cp{o+h)e% xtdT
fc=i
L a p l a c e t r a n s f o r m a t i o n 59
with an arbitary fixed a > 0 where 9 ?(o-\-ir) as a function of r is infinitely differentiable and of compact carrier. Then
.
( L t ) ( c p ) =2тcT [(Z -V n,
w h e r e { L ~ l q>)v { t )
= (
L ~ l < p ) ( — t).
Let us remark that L ” 1 is the inverse of the Laplace transform of L (see [1], p. 107).
Proof. We may limit ourselves to T = z^ , where zeL M(0, 00 ).
Pirst, we prove the local integrability of (LT) (s) with respect to r, where s = ( 7 +гт. Let be any finite interval, then
Hence
J |L T (s)|dr< f \z{t)\e~at{§ \s\kdx}dt
a
0a
00 00 b
< f M [z(t)]dt + J N [e~atf |s|*dr] dt‘ < oc.
(LT){<p) = J LT{s)<p(s)dr
— 00
00 00
= j г(£)| J ske~st(p(s)dT^ dt
0 — OO
00
= zt j j" sk e~stq>(s)dx j
— 00
( к 00
=<“ s )4 ( ^ — 00
00
= T,{ J e-“<p(s)dz} = 2жТ,[(Г-'<рГ].
— 00
Let us still remark that the same results are obtained if we consider the space T)'L 1 in place of D'M.
References
