1 0 1 00 0 +00

Download (0)

Pełen tekst

(1)

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

emma

1. 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 ) ,

(2)

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

he

ОКЕМ 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

heorem

2. 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

> - { - OO

ОС

— > — OO

exist, 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)

(3)

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)

(4)

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))

= 0

a—>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.

(5)

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.

Obraz

Updating...

Cytaty

Powiązane tematy :