ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X (1966)
ANNALES SOCIETATIS MATHEMATICAE POLONAE Serio I: COMMENTATIONES MATHEMATICAE X (1966)
M. Marjanovic (Beograd)
On a limit in the theory of distributions
1. Introduction. For a distribution f{x), —o o < x < + 00 limits
limf ( a x + x 0), lim f{x+(5) a— >0 f}— O O
the two
have been considered (see [1], [2], [4]). The former, if exists, means the value of a distribution f {x) at the point x — x 0 , and the latter is the value at x = ± 00. The aim of this paper is to investigate the following two limits
lim f(ax), lim / ( ax), where f(x) is a given distribution.
2. First, we are going to give a slight modification of Zielezny’ s lemnia (see [2]).
L
emma1. I f a distribution f(x), — о о < ж < + °°> satisfies the equation
f{Xx) = f (x) for every Я > 0, then it is of the form
f (x) — aH(x)-\-bII{— x), where a, b are constants and H(x) is
i f и = I®
P r o o f. From
Heaviside's function for x < 0,
for x > 0.
f ( x ) = lim
a—>0
f ( x + a x ) —f(x)
ax ’ x Ф 0,
it follows that f ( x ) = 0, for x Ф 0. The distribution/'(r), having a sup
porting set consisting of only one point, the origin, may be written as:
f ( x ) = а 0 д { х ) + а 1 д' { x ) f - ... + a k_ xd(li- l){x), where aj, j = 0, . .. , Jc— 1, are constants. Thus,
f ( x) = e-\-aQH (x )f-a 1 d(x) + ... -f a ^ d ^ i x ) ,
10 M. M a r j a no v i c
where c is another constant. Substituting this into the equation/(Аж) = /(ж ), we get
f{Xx) - f ( x )
= a [ H { X x ) - H ( x ) ] + a 1 ^ ^ - - l j ó ( x ) + ... -\-ak^1 l j <3(fc_1)(a?) = 0.
For every A > 0, H( ±Аж) = H( ± x ) and therefore a 1 = ... == ak_ x = 0, while a 0 is arbitrary. Finally,
f{x) = c + щ H(x)
or, taking into account that c = cH (x)Jr c H ( —x), we get the form f (x) = a H ( x ) + b H ( - x ) .
T
heОКЕМ 1. I f the limit lim /(аж) exists, then it is of the form :
Cl—^ -f- OO
а Н { х ) + Ъ Н { - х ) .
P r o o f. Let us put g(x) = lim /(аж). Then, we have
a—>+00
g (Аж) = lim /(аАж) — Нт/(аж) = g(x) for every A > 0,
a—> + 00 a—>00
and in view of the preceding lemma
I Пт/(аж) = аН(х)-\-ЪН(—х).
« —>-00
Note that
lim /(аж) = lim / ( — аж) = ЬН(х)-\-аН( — ж),
а— oo а—>-+00
and it is enough to consider only the case when а —> + 00.
T
heorem2. I f f(x) is a distribution of bounded support, then Ит/(аж) = 0.
a-^-oo
I f /(ж) is a locally integrable function and if the limits lim /(ж), lim /(ж) ОС
> - { - OOОС
— > — OOexist, then Ит/(аж) also exists and
a—>00
Нт/(аж) = lim /(ж) Я (ж) + lim /(ж) H( —ж).
а—уоо X—> + оо ОС—у— оо
P r o o f. Let f(x) be a distribution of bounded support, then there are continuous functions F a{x) of bounded support too, such that
/(* ) = 2 FfHce)
On a limit in the theory of distributions 11
(see [3]). For every (p(x)el), we have (f(ax), cp{x)) = |/(ж), % j
\(7<:v
q ^ p — со
where ( . . . , . . . ) are the duality brackets. Now, we have to prove that every term under the sign £ tends to zero as a ->■ oo. But it follows immediately from
+ oo
/ ^ 7 * ) -тлпУ”’ (“ -)*>!
< — jm ax \Fa{x)| max \<p(,1) (x) | mes {support of .Fg(a?)}^0.
cr + Hence, Ит/(аж) = 0.
а—>оо
Let now f (x) be a locally integrable function. Putting f (x) = f - ( x ) + f Q( x ) + f + (x),
where / - (я)
/(>')• x e( oo, - 1 ) ,
0, otherwise, / + И =
f(x), fl?e(l, + oo), 0, otherwise,
/о И
f (x) , я?е[ —1, +1], 0, otherwise,
we obtain for f 0 (x), as a function of bounded support, И т /0(аж) = 0. It
remains to be proved that a_M3°
lim /+ (ax) = lim f(x)H(x).
a—у oo X—y+oo
Given £ > 0, let a = lim f (x) and let X (e) be such a number that
ж -> + оо
£ £ + °°
a ---- — </(a?) < u + -^r- for a? > ,X(£); L = J |ę?(ir)|<?a?, (peD.
— CO
2L Then we get
(f + ( a x ) - aH (x ), <p(x))
2 L
I 1 / x\\ +0° i ix\ X(e) +0°
= l f + {x)—aH(x), —<p - = f (f+ ( x ) - a ) - < p (—\ dx = J + f .
\ а \ а / / 0 a \а/ 0 jf(e)
12 M. Ma r j a n o v i c
Since
\f+(x)-a\ — 9>( —I < (\f+(%)\ + \a\)ma,x\<p(x)\-,
1 a \ a
by the Lebesgue theorem on dominated convergence,
X f \ \ 1 l X \ 7
[f+ ( x ) - a ) - ( p l - \ d x
x a \ a! > a > a 0 (e)
and
-f-oo -fo o
[ / (/+ n - « ) f f g )
1 X ( f ) ' ' А » V '
I dx <
From this it follows that
И т ( / + ( а ж ) -
aH(x), <p(x))
= 0a—>oo
or else
lim f + (ax) — aH (x).
a—>0O
In the same way the equality
lim /_ (ax) = lim f ( x ) H ( —x)
а—>oo x—у— oo
is proved.
Theorem 3.
I f f (x) is a periodic distribution of period
2л:,then the limit lim / ( ax) exists and
a -> + o o
1 П Urn f(ax) = — - |/(*)ebc,
a -> + o o -*7C J
where the integral is taken in the sense given in [ 2].
P r o o f. There exists an integer к > 0 and a continuous function F(x), periodic of period 2
tc, such that
7T
f (x) = F {k)(x) + - i - J f (x) dx
— 7T
(see [2], p. 46, Lemma 20.3).
But F(ax)/ak tends uniformly to zero as
a -> oo.Hence we obtain lim F ^ ( a x ) = 0
a - » - f o o
and, by the above equality, the desired result.
I wish to thank Dr. Z. Zieleźny for his improvement of the text.
He also noticed that the limit in Theorem 3 must exist and the above
proof is his own.
On a limit in the theory o f distributions 13
References
[1] S. Ł o j a s i e w i c z , Sur la valeur et la limite d'une distribution dans un point Studia Math. 16 (1957), pp. 1 -3 6 .
[2] J. M ik u s iń s k i and R. S ik o r s k i, Elementary theory of distributions I Rozprawy Matematyczne 12 (1957).
[3] L. S c h w a r t z , Theorie des distributions, Paris 1950-51.
[4] Z. Z i e l e ź n y , Sur la definition de Łojasiewicz de la valeur d'une distribution dans un point, Bull. Polon. Acad. Sc., Cl. I l l , 3 (1955), pp. 519-520.
