JOSE SEBASTIAO E SILVA
TEXTOS DIDAcTICOS
Volume III
SERVI<;O DE EDUCA<;AO E BOLSAS
FUNDA<;AO CALOUSTE GULBENKIAN
I
LISBOA
Reservados todos os direitos de acordo corn a lei Edic;ao da FVNDA<;AO CALOUSTE GULBENKIAN Av. de Berna - Lisboa
1999 ISBN 972-31-0971-9 Dep6sito Legal
n.O
148805100
111.1 THEORY OF DISTRIBUTIONS路
*
Este texto tern por base apontamentos coligidos por diversos alunos de Jose SebastHio e
Silva na sequencia de urn curso que realizou em
1958 na Universidade de Maryland, e que
posteriormente foram utilizados, e por ele revistos, na Faculdade de Ciencias de Lisboa.
CHAPTE R
VI
LI M ITS A N D I NTEG RALS O F DISTR I B UTIO N S 6.1. Limits of a distribution as x � + 00 Let I be an open interval unbounded on the right; i.e. of the form 1 = ] a, + 00 [ with
a
E IR U { 00 } . The following two definitions are -
well known in classical analysis .
6.1 . 1 . DEFINITIONS. Let f and qJ be two functions on I. The func
tion f is said to be of order less than tp iff 3 Xo E IR and a function
f0 such that:
f = qJfo for x >x (i) and fo (x) � O as x � + oo. On the other hand, f i s said to b e at most of the order of qJ as x�+oo
iff 3 Xo E IR and a function f0 bounded for x
>x
0 '
such that f = qJf0 •
In the first case we shall write:
f E o (qJ) as x � + oo (or, on the right) and in the second case :
f E O (qJ) as x � + oo (or, on the right) .
86
These relations replace the classical f = o (cp) and f = O ( cp) which are not logically correct and may produce confusion in functional analysis . Observe that: 6.1 .2. If there
exists Xo such that cp(x) � O for x>xo' then
f E o ( cp) as x � + oo
f E O ( cp ) as x � + oo In order to extend where
"0"
�
�
f(x) � O, as x � + oo cp (x)
f(x) . . h t. lS b oun de d on th e rzg cp (x)
to distributions, we first consider the case
cp=i a, with a > - l (for simplicity the sign
"A"
Let be � the Lebesgue integral operator defined by in I.
will be omitted) .
ff@ d,; C
with
c
a is a real number > - 1 and f a continuous function such that f E o (xa ) as x � + oo, then �f E o(xa + l ) as x � + oo. PROOF. S uppose f E o (xa) as x � + oo. This means that there exists Xo and fo such that f =x a fo for x>xo and fo� O as x � + oo. Let 8> 0 b e given ; then 3 xl E IR such that I fo(x) I < 8 for all x>xl . We can as su me x! >xo > O. Now, for every x >x 1 6. 1 .3. LEMMA. If
;sf(x) = K+ Since I fo (x) I < 8 and
where K=
tf.
�> O, for �>x l ' we have:
�f(x) xa+ !
--
and therefore, since
f,; afo@ d,; I KI
s -- +
xa + !
a>-1:
l x a+ _ x 1 a+ 1 8 Vx >x I ' ( a + l )x a+ I '
87
�f(x) lim -+! Xa
x --+ 00
s
-
.
(';( f
As 8 is arbitrary, this implies that
�f(x) ! xa +
�
0
a+ 1
(
as x � + oo ; 1.e. ;S Eo x
a+l
).
.
6 . 1 . 4 . Remark. This lemma obviously extends to locally summable
fun ctions and even to measures, as we shall see. The lemma suggests the following :
Let a be a real numQer > - 1 and f a distribution on I. We write f E o (x a ) as x � + oo iff there exists an integer p � O and a continuous function F on I, such that: 6.1.5. DEFINITION.
f = DPF and
F(x) X
a +p
� O as x � + oo .
The lemma implies that if there exists p E /No and FE C(I) satisfying the preceding conditions, then every integer m > p and every function G such that G = � m -PF + P where P E CZPm , satisfies 6. 1 .6. Remark.
the same conditions (observe that if P E CZPm , then
6. 1 .7 . LINEARITY PROPERTY .
x � + oo , with a > - l , then:
0 (x) � as x � + oo) . + m X
P
a
If f Eo (xa ) and g E o (xa ) as
For the proof, it is sufficient to represent f and g as derivatives of the same order of continuous functions, taking into account 6.1.6.
88
In particular,
a may be equal to O . Then XO 1 and if f E o ( 1 ) as =
X ---+ + oo , it is natural to say that f---+ O as x ---+ + oo . More generally, let Il be any complex number and f E ďż˝(I) ; then : 6.1 .8. DEFINITION. We say that f converges to Il as x ---+ + oo if and only if f - Il E o ( l ) as x ---+ + oo . A distribution f is said to be converÂ
gent as
x ---+ + 00 if and only if 3 1l E CC such that f ---+ Il , as x ---+ + 00 .
Taking definition
Il = DP
( ) IlxP
pJ
6 . 1 . 5 . into account and observing that
for every p E /No , we can define the preceding concept
as follows :
6.1.9. DEFINITION. We say that f ---+ Il as x ---+ + 00 if and only if there exists p E /No and F E e(l) such that:
f DPF and =
F(x) xP
---+
-pJ Il
as x ---+ + 00 (in the ordinary sense) .
"f tends to Il as x ---+ + 00 " , we shall sometimes write "f (x) ---+ Il , as x ---+ + oo ", but it should be remembered that in these cases x is a dummy variable. Remark. Instead of
6. 1. 10. lf f ---+ Il as x ---+ + oo and f---+ J.1 as x ---+ + oo then 1l =J.1.
-
In fact, if f - Il ---+ 0 and f J.1 ---+ 0 as x ---+ + 00 , then, by
6.1.7.
( f- Il) - (f-J.1) = J.1 - 1l ---+ 0 as x ---+ + oo . But, for every integer p > 0 and every continuous function F such that J.1 - Il = DPF, we have necesxP
sarily F = ( J.1 - Il ) - + P where P E r;; .
pJ
P
Hence by definition 6 . 1 . 9 . , J.1 - 1l cannot tend to 0 unless 1l =J.1 . This makes legitimate the definition complementary to 6. 1 . 8 .
89
6. 1 . 1 1 DEFINITION. We say that A is the limit of f as x � + 00 , iff f � A as x � + OO . In this case, we shall write ..1, = Um f(x) or ..1, =f(+ oo) . x - + oo
The uniqueness of the limit is guaranteed in 6. 1 . 1 0, and from 6 . 1 . 7 . , follows : 6.1 . 12. LINEARITY PROPERTY. If f and g are convergent as
x � + oo , then:
Um (af + f3g) a Um f + f3 Um g , V a, f3 E C . =
x -- + oo
x - + oo
x - + c:o
In turn, from 6. 1 . 3 . and the preceding definitions, it follows : 6.1. 13 .
If f is a continuous function such that Um f (x) = A in the x --+ + oo
ordinary sense, then the same fact holds in the distributional sense; i. e. , in the sense of definitions 6. 1 . 1 1 . and 6. 1 . 9 . . Observe, that according to 6 . 1 . 5 . , this theorem extends to locally summable functions (and even to measures) . However, it -must be observed that the converse of this theorem is not true. 6. 1 . 14. Example. As is well-known, the function
cos x is not con
vergent in the ordinary sense as x � + oo . B ut we have:
lim cos x 0, in the distributional sense. =
To see that, it is enough to apply definition 6. 1 . 5 . observing that
cos x = D sin x and
Sln x x
�
O as x � + oo . ,
6. 1 . 1 5. General remark. All preceding definitions may be extended
and all propositions remain true, if we replace throughout + 00 by 00 and "on the right" by "on the left" . In particular, we must then con sider an interval I, unbounded on the left, 1= ] 00 , a [, instead of an interval unbounded on the right. -
-
90
6. 1. 16. DEFINITION. We say that f tends to A as x � 00 and we write Um f(X) = A if and only if Um f(x) = lim f (X) = A . x --+ + oo
x � oo
x --+ - oo
For example, it is easily seen that (cf. 6. 1 . 1 4) : Um cos x = 0 (in the x � oo
distributional sense) .
6.2. Limits and value of a distribution at a point of IR Let now I be any open interval ] a, b [, bounded on the left . Then, definitions 6. 1 . 1 . and 6. 1 .2. are readily extended to this case, replacing throughout "x � + 00 " by "x � a + " and "on the right" by "on the left." If we place � f(x) a
=
IXf(;) d;
, we prove, as for 6. 1 . 3 . (the proof is
even simpler) :
6.2. 1. LEMMA. If f is a continuous function on I, such that f E o [(x-a) p ] as x � a+ where /3> - 1 , then �:f E o [(x-a) P + Il ] as x � a+, for n = O, 1 , . . . . This lemma j ustifies the following
6.2.2. DEFINITION. If f E �(I) and /3> - 1 , we write f E o [(x-a) p ] as x � a+ iff there exists p E /No and F E C(I) such that f = DPF and
F(x) -- � 0 as x � a . --..,,(x-a) P +P +
6.2.3. Remark. The lemma implies that if there exist p and F satis fying these conditions, then every integer m � p, along with the func tion �am - pF, satisfies the same conditions. (But it must be observed that for each integer m > p, there is no function different from �am - pF satisfying the same conditions) .
91
Now we are able to extend definitions 6. 1 . 8 . and 6. 1 . 1 1 . , as well as prop ositions 6. 1 .7 . , 6. 1 . 1 0. , 6.1 . 1 2. and 6. 1 . 1 3 . , replacing + 00 by a + . In p articular, the convergence as x � a + can be defined directly as follows : A distribution f on I= ] a, b [ iff there exists p E /No and F E C(I), such that:
6 . 2 . 4 . DEFINITION. x
--:,
a
+
f = D PF and
-
A.. F(x) � (x - a)P pI
tends to
A..
as
+
as x � a (in the ordinary sense) .
Besides, the concepts of convergence corresponding to the cases x --:, + 00 and x � a + are related to each other according to the follow ing rule: 6. 2.5. Suppose
(
1= ] a, + 00 [, f3 > O and f E §}g(I). Then, if
)
g(t) = f a + f3 � , t
we have: Um
t -+ + oo
g (t) = A..
<=>
Um
x -+ a +
f(x) = A.. .
PROOF. This obviously reduces to the case a = O and
f3 =
1.
Suppose
f(x) � O
F E C(I) such that
f D PF =
x � O+ .
as
and
4.5), we have g<t) = <- t2D,) PF
Then, there exists p E /No and
F(x) �0 xP
(�)
A.. = O with
as
x�0
+
.
Moreover (cf.
and it is easily shown by induction
on p that there exists p + 1 numbers
ak
(whose expression are not
needed here) such that 6.2.6.
Now, since
F(x) xP
�
0 as x � O+, tPF
(-1 ) t
�
0 as t� + oo. Hence
92
which according to definition 6. 1 .9. means that all terms on the right side of 6 .2.6. � O as t � + oo . In a similar way, we prove that, if
g (t) � O as t � + oo, then f(x) � O as x � O+ . • We can obviously define the concept :
as we did for the case x � a+ considering now an interval ] a, b [, bounded on the right. It is readily seen that all preceding proposi tions and remarks can be extended to this case. Let i, be now any open interval in IR, i ] a, b [, and let c be any point of i, that is a < c < b. Then if f E liJ(I), we define the concepts : =
by considering, instead of f, its restrictions to the intervals ] a, c [ and ] c, b [. As in classical analysis, we shall put
f(a+) = lim f(x) (right-hand limit of f at a) x --+ a +
f(a- ) = lim f(x) (left-hand limit of f at a) x --+ a -
whenever the limit in question exists . 6.2.7. DEFINITION. We say that f tends to A as x � c iff f(x) �A as x � c+ and f(x) �A.. as x � c-. In this case, we write A= lim f(x).
According to preceding definitions and remarks, we can also define directly this concept:
93
6.2.8. DEFINITION. The distribution
f tends to A- as x � c iff there exi sts an integer p � 0 and a function F continuous at every point x of I distinct from c, such that: f = D PF and Urn x-c
F(x)
(x- c)P
=
A-
- in the ordinary sense.
p!
6.2.9. Remark. Suppose, more generally, that J is any non-degener
ate interval in /R and that c is in the closure of J. Then, definition 6.2. 8 . applies, even if c is a extremity of the domain J of f ; for ex ample, if c is a left extremity of J, we have by definition:
Urn f(x) = lirn f(x).
x-c
x-c+
With respect to the general hypothesis considered above, we have : 6.2.10. DEFINITION. A distribution f on J is said to be continuous at a point c iff there exists p E /No and F E C(J) such that f D PF and =
F(x) . . · --- IS convergent In the ord·Inary sense as x � c. Then, we wnte: P (x - c)
F(x) f(c) Urn f(x) =p! Urn -(x - c)P =
x-c
x-c
and the number f (c) is said to be the value of the distribution f at the point c (or, for x= c). From the linearity property of limits follows : 6.2.1 1. lf f and g are continuous (af + f3g) (c) = af (c) + f3g (c).
at c, so is af+f3g for a, f3 E C and 1
Examples. 1 - Consider f(x) = cos - . Then x function on /R and since:
f is
a locally summable
94
1
.
( 2 . 1)
1
cos - = 2x szn - -D x szn - , x x x
(
lim x sin �
x -+ O
X
)
=
0
,
it is easily seen that f is continuous at the point 0 with the value 0 (in distributional sense, and not in ordinary sense ! ) . 2 - It can be seen that at 0 for any k= O, 1 , . . . .
lim 8(k) = O,
x -+ O
-
k
and yet 8( ) i s not continuous
3 - It can be proved, as an exercise, that: If f is a distribution on an interval I minus a point c of I, and if f is convergent as x� c, then there exists one and only one distribution f on I U { c }, which is con�
tinuous at c and such that f = f on l.
-----
Remark. The previous concepts of limits and value of a distribution
at . a point of IR have been introduced by Loj asiewicz. As for the concepts of limit as x � + oo or as x � - 00, the definitions given by Mikusinski and Sikorski seem to be to restrictive as they are not invariant for very simple substitutions such as x = 1 1 t and do not allow the justification of certain integral fonnulas occuring in applications . The definitions that we are using here do not present these incon venIences .
6.3. Primitives and integrals of distributions
If f is a distribution with domain in IR, we call primitive of any distribution qJ such that DqJ f. From this definition follows:
f
=
Every distribution f has infinitely many primi tives, and, if the domain of f is an interval then any two primitives of f differ by a constant. 6.3.1. THEOREM.
95
PROOF. In the general case, the domain of f will be the union of a s ystem of mutually disj oint intervals (cf. 2 . 5 ) ; so we can reduce th i s to the case of a single interval. Let f be a distribution on I. Then f is of the form f = D nF, with F E C(I), and every distribution qJ of the form qJ=Dn:JF + K, where :J is an integration operator and K E C, is obviously a primitive of f. Suppose now that DqJI = D qJ2 ;... f ; th en if qJI = Dn(/)I and qJ2= Dnf/>2 ' with (/)1 and (/)2 in C(I), we have 1 V n+ 1 f/>I = Dn+ f/>2 ' which implies , by axiom 4 (cf. 2 . 2) , that f/>I - f/>2 is a po lyn omial P of degree < n + 1 . Thus qJ - qJ2 = D nP = constant. • I From 6 . 2 . 2 . and 6.2. 1 0. follows immediately :
If there exists a primitive of f which is con tinuous at a point a, then every primitive of f is continuous at a . If, in addition, the domain of f is an interval I, then for every complex number K, there exists one and only one primitive qJ of f such that 6.3.2. COROLLARY.
qJ (a) = K. It will be natural to denote by the symbol
f
f ( � ) d � or shortly by
ff
the primitive of f assuming the value 0 at a. (Remember that the sign A indicating that x is a dummy variable may be omitted whenever no confusion is possible) . Thus according to 6 . 3 . 2 . , if there exists at least one primitive of f which is continuous at a, the differential equation DqJ = f will have a single solution satisfying the initial con dition qJ (a) = K, and such a solution is :
As we have observed, it is understood that here x is only a dummy variable; the distribution qJ need not actually have a value qJ (x) at every point x of I. B ut, obviously, if qJ has a value at some point b of I, this value is naturally denoted by :
96
ff @ df ff (';) d'; ( ff ) rp (b ) = K+
Thus the integral
in short
is
defined
by
the
generalized B arrow Formula:
ff(x) dx = rp(b) - rp(a) . C oro l lary 6.3 .2. can be extended as follows :
6.3.3. COROLLARY. If there
exists a primitive of f having a limit as x --+ a + [resp. as x --+ a- ], then every primitive of f has a limit as x --+ a+ [resp. as x --+ a- ] . If, in addition, the domain off is an interval I, then for every complex number K, there exists one and only one primitive qJ of f such that qJ(a + ) = K [resp. qJ(a- ) = K] . Remember that the existence of both qJ(a + ) and qJ(a- ) does not imply the existence of qJ(a). All preceding remarks and conventions may now be extended to the newly considered cases. For example, we shall denote by
the primitive of f on I which tends to zero as x --+ a - ; accordingly, if such a limit exists, the differential equation DqJ= f along with the initial condition qJ(a - ) = K will have the only solution rp (x) = K+ So, we have by definition
f-f@ d'; .
97
If
b, the se are, respectively, the integral of the distribution f on the intervals [a, b [ and [a, b l . The integrals of f on la, b l and la, b [ are an alog ously defined. Naturally such an integral is said to exist or a<
to be c onvergent iff the two corresponding limits exist. If b s a, we
h ave of course :
Fin ally, all p rec eding definitions may be extended to infin ite tervals. For example, we have by definition
if qJ is a primitive of exist; and
f
a- f(x) dx L+oo
in足
such that the limits on the right-hand side is called the integral of
f
on the interval
[a, + 00 [ . For other kinds of infinite intervals the definitions are quite analogous . In the general case, a distribution f is said to be integrable over an interval I, iff the integral of f on I exists. This integral may be denoted by
L f (x) dx
or simply by
L f.
From the linearity property of limits follows immediately the corresponding property for integrals :
If two distributions f and g are integrable over I, so is af + pg for any a, PE C and
6.3.4. LINEARITY PROPERTY.
On
the other hand it should be observed that:
98
6.3.5.
If! is a function summable on I, then the integral of f over I,
in the distributional sense, exists and equals the Lebesgue integral over l. More generally, if f is a locally summable function on I such
L ! is convergent in the classical sense (even simply convergent), then L ! exists, in the distributional sense, and has the same value. that
However, the converse of this proposition is not true, as we shall presently see: Examples. 1
-
Consider the integral
interval in fR , n an integer
>
L ! (x) o<n\x - a)
where I is any
0 and f E e n a function on
l. Then:
Now, for every k < n, a primitive of D n -k[ f Ck) (a) 8(x- a)] is the distri bution f Ck)(a) 8(n -k- l ) (x_ a) which tends to zero as x tends �o any point Xo in fR. Hence:
L I
f(x) 8 Cn) (x- a) dx = (-l ) nf Cn) (a)
L I
(- l ff Cn ) (a) , { 8(x-a) dx = .
if a E I
0
, 1f a � l.
For example:
af- +f(x) 8" (x-a) dx = f " (a) fa- + 8(x - a) dx Ja Ja JfIRei {J)tdt,
2 - Consider the integral
where
0)
=
f " (a) .
is a real parameter. This
integral is obviously divergent, in the classical sense, for every value of
0).
However, for
0) �
0, one primitive of e
i {J)t
i
is
wt
� l O)
and
99
--
tan
1 e i wt ' De i m (im) 2
(J)(
ei lim - = 0. ( -+ 00 t
Hence, we have, in the distributional sense, for every
m ďż˝ 0:
1+00ei {J)tdt - -im ( (I¡-+lm ei {J)t-(I¡-+lm00 ei {J)( ) -- 0. 1
- 00
For
+ 00
m = O, this integral is divergent,
-
even in the distributional sense.
This result agree with the intuition of physicists, which have, long since, adopted the formula:
1 ei {J)(dt = 2iro(m). IR
However, a complete justification of this formula cannot be achieved, without a suitable definition of parametric integral, which will be given in chapter VIII. The case considered in example 1 is included in the following proposition : 6.3.6.
Every distribution with a bounded carrier on fR is integrable
on fR . PROOF.
Let
f
be a distribution of bounded carrier on fR .
1= [a, b] such that f=O outside /. Hence, if cp is a primitive of f, Dcp= O outside I and cp reduces to constants C l and c2 ' respectively, on ] - 00 , a [ and on ] b, + 00 [. Thus cp(- oo) = cp(a- ) = c I and cp(b+ ) = cp( + 00 ) = c2 . Hence, f is This means that there exists a bounded interval
integrable on fR and
1 00
A complementary proposition to 6.3 .6., which can be proved in
a similar way is the following :
Whenever f is integrable on /R , we have r f JIR . . . lnterva I eontalnlng the earner 0if f .
6.3.7. .
For ex ampl e , if f i s integrabl� on /R and
r f= JIR
=
r f, jor every J I
zero fo r
l� f.
x<
a, then
oo
a
In order to obtain more powerful tests for the convergence of integrals, we are going to develop the concept of order of growth for distributions. 6.4. Orders of growth for distributions
For brevity, we shall confine ourselves to the typical case where x � + 00, since the considerations in the other cases are analogous. Let [ be any interval
unbounded on the right and �/(x) =
e El, for fE C(/). The extension of the symbol
f/,
with
"0" to distributions
is based on the following lemma, whose proof is similar to the one of 6. 1 .3 . and even more simple: 6.4. 1. LEMMA. Iff is a
eontinuousfunetion on I sueh that f E O(x a ) as x � + oo, with a>-I , then �fEO(x a + ' ) as x � + oo. 6.4.2. DEFINITION. If f E §(/) and
as x � + oo iff there exist
n E /No and F E C(/),
F(x) . . IS bounded on the nght. xn+ a
--
a > -I , then we write f E O (x a ) such that
f= DPF and
101 particular:
Th e lem ma guarantees the linearity property for this case . In
6.4.3. DEFINITION. A distribution f on I is said to be bounded on the right iff f E O( 1 ) as x -+ + 00 , that is iff there exist n E 1No and F E C(I) such that f D F and =
n
F(x) . . IS bounded on the nght. n x
That being so, we are able to define the meaning of the ex
pres si on "f Eo (cp)" and "f E O (cp)" in the more general case when f E !0"(I) and cp E C OO(I) . For all that purpose, we can take as a model
the cla ssical definition 6
.1.1.
:
6. 4.4. DEFINITION. We shall write f E o (cp) as x -+ + oo iff there exists a real Xo and a distribution f0 such that: f = cpfo for x >xo and fo -+ 0 as x -+ + oo .
shall write f E O (cp) as x -+ + oo iff there exists a real Xo and a dis tribution f0 such that f = cpf0 for x > Xo and f0 is bounded on the right.
We
The first thing to do is to see whether these definitions are equivalent to the preceding ones in the particular case, when cp is of the form x a, with a >-l . This equivalence is easily proved by means of the formulas :
x aD nFo =
�o <_ 1 )k (:)
D n - , ( FoDx'xa )
taking into account the linear property. On the other hand, this same property can be now immediately extended to the general case. Moreover definition 6.4.4. introduce a remarkable new property which is a counterpart of the preceding lemmas .
1 02
6.4.5. DIFFERENTIATION PROPERTY. If f E O(x a) on the
right,
then Df E O(x a - l ) on the right, for every a E /R . We shall begin the proof in the case
a= O:
6.4.6. Iff is bounded on the right, then Df E O(x - l ) as x � + oo. Suppose f bounded on the right. Then, there exists p E /No ' F E C(J) and c such that f DPF for x > c and =
right. We may choose
F(x) xP
is bounded on the
c > 0; then we have:
and it is readily seen that DP + I (xF) is bounded oil the right, as well as DPF. Hence Df E O(x- l ) as x � + oo. Suppose now fEO(xa) x � + oo, where a E /R . Then there exist Xo and fo such that f =xa fo for x >xo and fo E O( I ) on the right. It fol lows that Df axa - 1fo +xa Dfo and it is readily seen, applying 6.4.6. , that Df E O(xa - l) as x � + oo . • =
B y an identical argument, it is shown that the
property extends to the
"0 "
differantiation
symbol.
Furthermore it is a simple matter to prove the following prop erties where the expression "on the right" or "as x � + 00 " is omitted for simplicity.
6.4.7.
If f is convergent, then f is bounded.
6.4.8.
If f Eo(cp) then f E O(cp).
6.4.9.
If f E O(xa) and a< /3, then f Eo(xfJ ).
Obviously w e have chosen the case when x � + 00 a s a model ; the concepts and properties are quite analogous in cases such as x � - oo, x � c + , etc . .
1 03
6 . 4. 1 0 . Convention. If a distribution
j has the same growth property shall say that j has this property as
x � + oo and as x � - oo, we x � oo. If j E PLJ(I) is bounded on the right and on the left (respec tively as x tends to the right extremity and to the left extremity of I), w e shall say that j is bounded o n I or simply bounded . as
The concept of bounded distribution that we have just in troduced is more general than the concept of bounded distribution accordi ng to Schwartz and necessary for the integral theory as we
Remark.
shall next see.
6.5. Convergence tests for integrals
Let us consider, at first, the case of integrals on fR . We have the following test, which is not true in classical analysis : 6.5. 1 . (A NECESSARY CONDITION FOR CONVERGENCE).
If a
distribution j is integrable on fR , then j E O(x- 1 ) as x � oo. PROOF. Suppose there exists a primitive cp 'of j such that
cp is convergent as x � + oo and as X � _ OO(6). Then by 6.4.7 . , cp is bounded on fR and, by 6.4 . 6 . ( and its analog for the case x � oo ) we have \
Dcp E O(x - 1 ) as x � oo . •
-
The following theorem extends to distributions a well known classical test. 6 .5.2. (A SUFFICIENT CONDITION FOR CONVERGENCE).
If there exists a number a < - l such that j E O(x a ) as x� oo, then j is integrable on fR .
(6) different.
-
This does not mean that qJ is convergent as
x-
00,
for the limits are in general
1 04
Suppose f E O(x a ) as x -+ oo with a < - l . Then there exists a number c > 0, an integer n � 0 and a continuous function F PROOF.
such that:
f = x a D nF for I x l > c, with
F(:) bounded for I x l > c. x
Set:
Then f2 is a distribution with carrier contained in [- c, + c] ; hence integrable on IR (cf. 6.3 .6.). So we have only to prove that fl is inte grable on IR , for then we have:
We shall put fl f and � F. Then : =
=
f = x a D nF
k (-l lck Dn - k (x a - kF )
where ck= a (a-l ) . . . (a-k+ l )
n
(�)
From here we deduce the follow-
ing primitive of f:
But since F E O (x ·n ) as x -+ oo in the ordinary sense, we have � a -nF E O( � a ) as x -+ oo with a < - l and, according to the classical
test, this implies that the primitive � a -nF is summable on lR. Hence the last term in 6.5 . 3 . is convergent as x -+ + oo and as x -+ - oo .
105 As to the other terms, observe that the functions
x a - kF (x) a + 1 F (x) for k = O, . . . , n - l = X n xn k- I x F (x) tend to zero as x � oo since a + l < O and is bounded (in the n ,
ordinary sense). Hence, by definition 6. 1 .9. Dn
-k- I
x
(X a-kF ) � 0 as x � oo,
Therefore f is integrable on fR and
We can deduce similar tests for integrals on intervals distinct from fR. For example, consider an interval 1= ] a + oo L and f E 2J(I). Then it i s easily seen that if f is integrable in I, then f E O(x-I ) as x � + oo and f E O( x - af' ) as x � a+. If there exists a < - l and ,
f3 > -1 such that f E O(xa) as x � + oo and f E O ( x - a)f3 ) as x � a+,
then f is integrable on I .
6.6. Multiplication and change of variables in connection with limits and integrals
It is a simple matter to prove the following propositions:
is convergent as x � c+ with c E I and if g E C OO(I) then fg is convergent as x � c+ and :
6.6 .1. lf f E 2J(I)
1 06
6.6.2.
If f E �(/) is convergent as x � c+ with e E l and if h is a C oo
mapping of an interval 1* into I, such that h ' (t» O in 1*, then f (h (t») is convergent as t � y + with h (y ) = c and: Um f(h (t») = Um f(x) .
t -+ y +
( -- c +
Obviously, these two propositions can be extended to the case when f is convergent as x � c- . Then the second one enable the usual substitution property to be extended to the integrals of distributions on bounded intervals. For example, assuming f E �(/), a, b E l and h is an increasing C oo mapping of 1* into I such that a = h (a), b = h ( [3 ), we have rr
L f(x) dx = L f(h (t»)h'(t)dt b-
whenever the first integral exist. However these criterions are not sufficient in certain cases which occur in practice. Our next purpose is to introduce a stronger criterium than 6.6.2 . . For simplicity, we shall reduce our discussion to the case where x ----). + 00 and h ( + (0 ) + 00, which can be taken ' as a model for other cases. =
Let f E �(/), I unbounded on the right, and let h be a C oo mapping of an interval 1* into I such that h '(t) � O on 1* and h (t) � + 00 as t � + 00. Suppose that: (i) f is convergent as x � + oo (ii) h ' tends to a number c � O as t � + oo (in the ordinary sense) (iii) h(k) E o (r k + 1 ) as t � + oo (in the ordinary sense), for k > l . 6.6.3. THEOREM.
Then we have: Um f(h (t») Um f(x) ( -+ + 00 x -+ + oo PROOF. Suppose f �A as x � + oo . =
and
F E C(/)
such that
f=D
n
F and
F (x) xn
Then there exist
A
-+ -
n!
n E /No
as x � + oo . Now
1 07
f
0
h
=
( �, DJ'(F h) 0
and according to the hypothesis :
h (t) Um -- = lim h'(t) = c . r � + oo t � + oo t
Hence:
( )
6.6.3 ' .
F(h (t)) F(h (t)) h (t) = n (h (t)r · t t
n �
ILcn . n!
On the other hand it is easily seen that:
where
ao =
( �, r
and
a E o (r ) as t � + oo , for k = l , 2, . . . , n. Thus ,
'
all terms in the last sum tend to zero as x � + oo, except Dr\ao(F o h)), which, by 6.6.3 ' , tends to IL . • This criterium and the corresponding ones for the cases when x � - oo , t � - oo, etc . , lead to the following substitution rule for in tegrals .
Let f be a distribution integrable on IR and h a C oo mapping of IR onto IR such that: (j) h'(t) is ;Z! O on IR and tends to numbers ;z! O as t � + oo and as t � - oo (in the ordinary sense) (jj) h(k) E o (r k + 1 ) as t � oo (in the ordinary sense) for all k= 2 , 3 , . . . . Then f (h (t)) is integrable on IR and: 6.6.4. COROLLARY.
f f(x) dx = f f(h (t )) l h '(t ) l dt. JIR JIR This rule is an immediate consequence of theorem 6 .6. 3 . and its
108
corresponding theorems applied to a primitive rp of f. Observe that, in the case h '(t) < O
oo oo 1:00f(x) dx = foo f(h (t» h '(t) dt = - [oo f(h (t» h '(t) dt .
In particular 6.6.4. applies in the elementary cases when x= t + a or x = e t, with a E IR and e E C. Then we have: 6.6.5.
1 f(x) dx = l e l l f(ex) dx .
6.6.6.
1 f(x + a) dx = l f(x) dx .
IR
IR
IR
IR
The last formula can be expressed by saying that the integral is invariant under translations. More refined criterions can be obtained by using the concept of measure as we did for multiplication in chapter IV. Remember that if /1 is a measure on an open interval I, the total variation of /1 in a bounded interval J such that J e I is defined to be the supremum of the sums Sp =
I into intervals
11 ,
•
•
•
,
� 1.u('!') I , p
for all finite partitions P of
Jp ' We shall denote by 1 /1 1 (J ) the total
variation of /1 in J; as is well known, 1 /1 1 is again a measure on IR (the modulus of /1) such that: (i) if /1 E t, then 1 /1 1 is the modulus of the function /1 in the ordinary sense; (ii) I rp/1 1 = I Jl I for all rp E C(/ ) . On the other hand, if /1 and v are two measures on I, we write /1 < v iff /1 (l ) s v(l ) for all bounded intervals I such that J e /. Suppose I i s unbounded on the right. A measure /1 on I i s said to be bounded on the right if and only if there exist two numbers Xo and k such that I Jl I < k for x > xo ; i.e. I Jl I (1 ) s k I 1 1 for all bounded intervals le [x o ' + 00 [ . On the other hand, we say that /1 converges to
rp l l
1 09
number
c as x ---;;. + oo, iff for every £> 0, there exists a real Xo such that L u- c 1 < £ for x > xo . It is readily seen that these concepts coin
a
cide with the classical ones if J.1 is a function. Besides, the preceding lemma for the " 0 " and "0" symbols keep true if f is a measure. These remarks suggest the following refinement of the concept of convergence for distributions: 6.6.7. DEFINITION.
Let f E �(/), n E /No and A E C. We write f ----;;+ A as x ---;;. + 00 if and only if there exist a real X o and a measure F
F (x) A ---; ;. (in measure sense) as x ---;;. + oo. On n n! x
n
such that f = D F and
the other hand, if cP E C (I), we shall write f E on ( cp ) as x---;;. + oo iff there exists Xo and fo such that f = cP fo for x > X o and fo ----;;+ 0 as x ---;;. + 00 . 00
The expression "f E On (cp)" can be analogously defined and the "dual" concepts of the preceding ones can be introduced as follows: Let n E /No ' f E C \/ ) and A E C. We shall write f � A as x + 00 iff f tends to A and f(k) E 0 (x - k' ) as x ---;;. + 00 , n for k = 1 , . . , n (in the ordinary sense). We write f E o (lp) as x ---;;. + oo iff there exists Xo and fo such that f = cP fo fo r x > X o and fo � 0 as 6.6.8. DEFINITION. ---;;.
.
x ---;;. + 00
•
Thus, it is readily seen that: 6.6.9. x ---;;.
If f ----;;+ A as x ---;;. + 00 and g � J.1 as x ---;;. + 00, then fg ----;;+ AJ.1 as
+ 00
6.6. 10.
•
If f ----;;+ A as x ---;;. + oo and if h is a
1* into I such that h C ;l!
00,
then f 0 h ----;;+ A as t
6.6. 1 1 .
---;;.
+ 00
---;;.
as t
+
---;;.
+ 00
00 .
n
Cn mapping
of an interval and h' � c, with C ;l! 0 and
If f E on (cp) and g E o (lf/) on the right, then fg E o n (cplf/) on
the right, and analogously for the
"0 "
symbol.
1 10
6.7. Scalar products. Definition of distributions according to Sobolev-Schwartz
We shall say the two distributions f and g are multipliable if and only if the product fg exists in some of the senses considered in chapter IV. That being so :
6.7. 1. DEFINITION. If two distributions f and g on an interval I are multipliable and fg is integrable on symmetrical scalar product
g and is denoted b y ( f, g ) :
/, then
L
fg will be called the
or simply the scalar product of f by
Obviously, the scalar product is in fact symmetrical (or com足 mutative) . Moreover it is bilinear: for all A, /l E e, we have
whenever ( f1 , g ) and ( f2 , g ) exist and analogously on the right. Observe that every distribution f on / can be written in the form f = u + iv, where u, v are real-valued distributions (i .e. , of the form u = D n U, v = D n V, where U and V are real-valued continuous func足 tions on I ) . Then we put f = u - iv (conjugate of f ) . It is readily seen that if ( f, g ) exists, then ( f, g ) exists also . We call product
L jg
the hermitic scalar product or simply the hermitic
of f by g, and it is denoted by
(f I g) :
The hermitic product is not commutative:
111
but it is of course, linear on the left.
R emember that any function
f E L2 (I)
(square summable func足
I) is locally summable, hence a distribution. It is well known that if f, g E L2 (I), then fg E L (I) so that ( f i g ) is a hermitic form on L2 (l), which makes L2 (I) a Hilbert space. The following is a classical tion on
theorem in functional analysis.
6 . 7.2 .
If E is a Hitbert space, there is a one-to-one correspondence
between the continuous linear functionals on E and the elements of E. The functional U corresponding to an element u of E is given by the formu la:
U(x) = ( x l u ) for all x E E . E and E ' . B ut the elements of E ' (covariant vectors) do not behave
Moreover, this correspondence is a vector isomorphism between
like the elements of E (contravariant vectors) by change of bases ; thus it is not convenient in most cases to identify E ' with E.
For developing the study of scalar distributions, a remark about terminology is necessary. When
I is
a compact interval, the expres足
sion "measure on I " is commonly used with a meaning equivalent to that of "measure of an interval contained in I " . For example, in this sense, 8 may be considered as a measure on
1= [0,
1 ] ; but the re足
striction of the 8 distribution to [ 0, 1 ] is D ( PI H ) = O. To avoid con足 fusion, we shall say "measure in I " instead of "measure on I " for a distribution f of the form f = DF, when F is a standardized function
I.
M* (I) the vector space of all measures on IR which vanish outside I of bounded variation on
On the other hand, we shall denote by
and by M (I) the set of all measures on I. Observe that:
b] , every measure f.1 in I can be uniquely extended as a measure l1EM* (I) such that 11 [a, a] = l1 [b, b] = 0. 6. 7.3. If 1= [ a
,
1 12
In fact, if f.1 E M(/), then f.1 = DF where F is a function of bounded variation on I. Now F can be uniquely extended to a func tion F of bounded variation on IR such that F(x) = F(a+ ) for x < a and F(x) = F(b- ) for x > b . .......,
.
_
-
-
-
------
� - - F(b-) --b
a
Hence, if we put ji= DF, we have jiE M* (/), ji = [a, a] = = ji [b, b] = 0, and it is readily seen that ji is uniquely determined by f.1 . We call ji the minimal extension of f.1 to IR. Another classical theorem in functional analysis is the following:
6.7.4. F. RIESZ THEOREM. There is a one-to-one correspondence between the measures f E M* (/) and the continuous linear function als u on C(/). This correspondence f u is given by: �
We are going to deduce some important consequences from this formula. We shall denote by M: (I) the set of all distributions of or der s n on IR vanishing outside I. Suppose 1= [a, b] ; then
6.7.5. Every distribution f E M: (/) can be written in the form: n-l
f= D nFo +koc,o(')(x - a)
where Fo E M * (/ ) and co ' . . , cn _ 1 E C. .
f E M: (/). Then f is of the form f = D nF where F E M(lR). On the other hand, since f = O outside I, F reduces PROOF. Consider
1 13
to p olynomials P and p* of degree < n, respectively on the left and p then ji ° for x > b and f= D n . We P*; ji on the right of I. Put F ,......, can suppose that F= F = O for x > h. Set:
po --
{p
for x < a
° for
Fo =
X "2! a
{o
F
for x < a
for X "2! a .
Then F= Fo + Po and FoEM*(I). On the other hand, � is a poly nomial of degree < n for x < o and zero for x > O, so that D n� is of the form:
Consequently n-l
f = D "Fo +"J-o c o(k)(x - a). k
•
Let now ffJ be any e n function on IR and f E M: (I). Then, of course, ffJf EM: (I); so that ffJf is integrable on IR . Put
f=D
n-l
o k OCk)(x - a) with j1 EM*(I) .
j1 + "J- C
"
A primitive of ffJD nJ1 will then be:
and since J1= ° outside
I,
we have:
On the other hand (cf. 6.3 .6. and example 1 in 6 . 3 . )
( ffJ, 8(k) (x - a») = (- 1 ) k ffJ (k) (a ).
1 14
Hence:
( t, q»
6.7.6.
�(_ l )kCk q>(k)(a) + (- l )"L q>(n)p .
Observe that in this formula the values of ([J (x ) for x tt l do not matter. So we may extend this formula by definition to all functions ([J In it is common to define the norm by :
EC\/). C\/)
1I ([J ll n= sup { 1 ([J (x ) l , I ([J '(x ) l , . . . , 1 ([J (n )(x ) l } .
6.7.7.
xEl
C n(/)
Then becomes a B anach space and the convergence of a sequence (([J ) to 0 in this norm means the convergence of the n se p quences (([J )' (([J� I ) , . . , (([J�n) to 0 uniformly on I. Now we have the p following consequence of the Riesz theorem: .
There is an isomorp hism f � u between the vec tor spaces M: (I) and C n (I)' defined by u (([J) = ( f, ([J) \;f ([J E C n (I) . PROOF. a) Take f E M: (I). Then f is of the form:
6.7.8. THEOREM.
/=D np +k Ck 8k) (x - a) n-l
o
with J1 E M* (I) and:
u (q» =
�(_l )'q>(k)(a)
+ (- 1 ) "
This defines clearly a linear functional U on we have:
L rp (n)p .
C n(I ) . On the other hand,
n-l
l u ( rp)1 s � lek l ll rpl l n+ Il rp lln l p l (I ) which shows that u (([J) � O as ([J � O. b) Take u E C n (/)' and set, for all f// E C(/), with ,;J lfI(x) =
f lfI(�) d� .
v(f// ) = (-l ru(,� n f// ) ,
Then v is a continuous linear functional on
1 15
C(/) and by 6.7 . 3 . , there exists j.1 E M* (I) such that v ( f// ) = ( j.1 , Besides , every function rp E C \I) can be written in the form:
Hence , if we set
U
s i nce
(
-a ck= (- l ) k u x t k! (ffJ) =
)
f// ) .
we find:
�(-1 )'c, ffJ(')(a) + (- I )"L ffJ (n)j.1
u ( ,J' n rp (n» ) = (-l f v (rp (n» ) = (- l f ( u , rp (n» ) .
rp E C n(I) .
•
So if we put
n- l
f = � c,O(')(x - a) + D nj.1 ,
we have
u (ffJ) = ( t , ffJ )
for all
We shall denote by C;(I) the set of all C n functions rp on I such that rp (k)(a) = rp(k)(b) = 0 for k= O, . . , n . It is obvious that C;(I) is a vector subspace of C n(I). Also, every rp E C;(I) can be uniquely ex tended as a C n function on IR vanishing outside I, so that C;(I) can also be identified with a subspace of C \IR). We shall consider C; (I) provided with the norm 1 1 · ll n defined by 6 .7 . 7 . On the other hand M (I) is the vector space of all distributions 1 on I of the form I =n D nj.1 with j.1 E M(I). Now from 6.7 . 8 . , follows:
.
There is an isomorphism I � g between M (I) n and C;(I)' defined by u ( rp) = ( /, <p) for all rp E C;(I). Besides ( I , rp ) = (-l r ( j.1 , rp (n» ) if I = D nj.1 . PROOF. a) Take I = D nj.1 where j.1 E M (/ ) . Then if we set J DnJi where Ji is the minimal extension of j.1 to IR (cf. 6.7 . 3 .), 1 defines a functional JiE C \I), whose restriction to C; (I) is obvi ously an element u of C; (I)' such that 6.7.9. THEOREM.
1 16
But as Jl [a, a] = Jl [b, b] = 0, it is easily seen that
So, we can write u (qJ) = ( f, qJ) = (- l r ( u, qJ(n » ) . b) Take now u EC* (J) ' . Observe that for every function qJ E C n(/ ) there is one and only one function qJo E C* (/ ) such that: n
tpo (x) = tp (x) -�a/x - a)k- (x - a r
� b.(x - b)k, n
where the coefficients a and b can be obtained as linear combina k k tions of the values qJ(k)(a) , qJ(k)(b) for k = O, 1 , . . . , n. We shall denote by 1C the mapping qJ � qJo of C n (/ ) onto C; (/). Since the a , b are k k linear combinations of the qJ(k)(a) , qJ(k)(b), it is readily seen that 1C is a projection, i.e. a linear mapping such that 1CqJo = qJo for all qJo in C; (/) and continuous . So if we set U(qJ) U (1CqJ ) , U will be a contin uous linear functional on C n (I) extending u ; hence there exists a dis tribution fEM; (/) such that u(qJ) = C f, qJ) and therefore., if we put f = p/ f, it follows that u (qJ) = ( f, g ). Finally suppose (f, qJ) = ( g , qJ) for all qJ E C; (J), with f= D nf,1 , g = D n v , v, J.l E M(J) . Then if we put f D n Jl , g= D n v , where Jl , v are the minimal extensions of f,1 , v, i t follows that (7, qJ) = (g, qJ) for all qJ E C n (J ) , s o that f g (by theorem 6 .7 . 8 .) and therefore f = g . • =
We shall now denote by C ; (/ ) the space of all C oo functions qJ on 1= [a, b] such that qJ(n)(a) = qJ(n)(b) = 0, for all n E /No . Such func tions can be identified with the C oo functions on /R with support con tained in l. In that space there is defined a topology making C oo an (F )-space by means of the sequence of norms 11 · l l n . This being so: 6.7. 10. THEOREM. There exists a vector isomorphism f� U tween lW(/) and C; (/) ' which is given by the formula u (qJ) = (f,
Besides
be qJ) .
1 17
6 .7. 1 1 .
(Df, cp ) = - u (cp ' ) , 'V cp E C; (/)
and, if J is a compact interval contained in I, then 6.7. 12. where
and J.
uJ
and fJ are the restrictions 0/ u and f respectively to C; (J)
PROOF. a) Take f E !?)J(/) . Then there exists an integer
n
such
that f = D nF with FE M(/). So if we set u (cp) = (f, cp) = (-l r ( F , cp) for all cp E C; (I),
u
is clearly a continuous linear functional on
C; (/). b) Take u E C; (/) ' . Now, a fundamental system of neighbor hoods of ° in C; (I) is given by the sets :
Hence, for any 8> 0, there exists an £ > 0 and n such that £u(BJ < 8. But this means that u is continuous with respect to the norm 1 1 · 11 n on C; (I) and we shall see in the next paragraph that C; (/) is dense in the normed space C* (/) . So u can be uniquely extended as a functional uE C* (/) ' ; i.e. there exists one and only one distribution f E M(/) such that u (cp) = (f, cp) . Finally 6.7. 1 1 . and 6.7 . 1 2. are easy consequences of the preced ing results . • This theorem shows that the dual space of C; (I) affords a model of the axiom system in 2.2. if we identify every function f E C (I) with the functional
u
such that
I
u (f{J) = f f{J and
if we define Du by
Du (cp) = - u (cp ' ) . As a matter of fact, Sobolev had first (in 1 936) the idea of taking such functionals as generalized /unctions (of real variables) . This method was developed in a systematic way by L. Schwartz in
1 18
1 944-45 . So, according to Schwartz the elements of C; (I)' are called distributions on I. The sum of two distributions u and v is the sum of the functionals u and v in the usual sense, the restriction of u to an interval l e l is the restriction of u to C; (l), and so forth. Till now, we have been concerned only with a compact interval l. Let us consider now an open interval Q in IR and let us denote by C; (Q) the set of all C oo functions on Q with bounded carrier, contained in Q. According to Schwartz, C; (Q) is provided with the topology obtained as the inductive limit of the topologies of the (F )-spaces C; (/). Then, a linear functional u on C; (Q) is continu ous if and only if the restriction u/ is continuous . This being so, Schwartz called the elements of C; (Q) ' distributions on Q. B ut now theorem 6.7 . 1 0. leads directly to the following : 6.7.13. COROLLARY. PJ (Q ) and C; (Q) ' .
There is a vector isomorphism f ++ u between
The result can obviously b e extended to any open set Q i n IR . It must be observed however that Schwartz denotes by PJ(Q) the space C; (Q) and by PJ ' ( Q ) the space of global distributions ·o n Q . On the other hand, S chwartz defines the topology on PJ (Q ) as the strong topology of C; (Q)' . But it can be proved without difficulty that this topology is the same one that we have defined directly in chapter V, i.e. the isomorphism in 6.7 . 1 0. is a topological isomor phism. The functional theory of distributions requires some warning in order to avoid misunderstandings . This begins already with mea sures . Observe, for example, that if f is a locally summable function on IR and f1 the corresponding measure, we have f1 (J ) =
i f (x) dx
for
every bounded interval I; but if we consider a one-to-one C l map ping h of IR on to IR , the transformed f.1* of J1 by h is given by
i
f1 * (1 ) = f( h ( t) ) h '( t)dt
1 19
so that J1 * is defined by ( f o h ) h ' , instead of by f o h Hence functions and m easures behave differently by change of variables, so that the identification offunctions with measures works only as far as a sub stitution x = h (t) with h ' � l is concerned. The same difference arises between global distributions (as we have defined them) and distributions according to Schwartz : the first behave l i ke functions and the second like measures, by change of variable. In such a situation, functions cannot be identified with line ar fu nctionals, since the first are contravariant vectors and the second covariant vectors. A s a last example, let us consider the space 5t \Q) of all functions f E C \Q) such that f (k) is square-summable on Q for k = O . . . , n (n E /No )' provided with the following definition of her mitic product: .
,
Then 5t\Q) is a Hilbert space, whose dual is just isomorphic to 5t n (Q). But according to Schwartz, 5t\Q)' is identified with the space
p
5tn(Q) of all distributions f of the form f= L D k vfv where p is v= l
an arbitrary integer � O and O � kv � n , fv E L2 (Q) for v = l , 2, Obviously this identification requires special care.
.
.
.
, r.
6.8. The approach of functions or distributions by means of C 00 functions. Distributions according to Mikusinski
Consider the function y defined as follows :
(
y (x) =
o
1
1 + exp
X )-1
x 2- I
for - 1 < x < 1 for for
x<- 1
x� 1
1 20
It can be seen by elementary calculations that y is a on IR increasing from 0 to 1 . Set
Hn (x) = y (nx)
6.8. 1.
Then
bn E C oo, �(x) = O
L: 8" (x)dx =
1 for
� = H:
and
n = l , 2, . . .
for
n = l , 2, . . . .
1
1 if I x l < - and
I x I � - , �(x» O n
if
C oo function
n
n -'i> ° (cf. 5 . 3 . ) .
. It follows that o
Moreover: , 6.8.2. LEMMA.
qJn (X) =
If f E C(/R) and
L:oo 0n (x - t) f(t ) dt for
then fPn E C OO(/R) for all n and interval . . PROOF. SInce
bn (x - t) = O
fPn � f
for
x E IR, n = l ,
2,
. . .
,
uniformly on each compact 1
Ix-tl >- , n
121
x
x- l In
x+ lIn
we have
tpn (X) =
r+l! 8,, (x - t) f(t) dt , n
'fix E IR ,
n = l , 2, . . . .
Consider now a compact interval J [a, b] and put Then: =
tpn (X) =
L 8" (x - t) f(t) dt ,
K
=
[a - 1 , b + 1 ] .
'r/x E J, n = 1 , 2, . . .
and since 8n E C (lR) for all n, it is readily seen that ({In E C (/R) for all n. On the other hand, by the mean value theorem there exists for 00
00
1
each x E J and each n 1 , 2, . . . a real ; such that I x - ; I < - and =
tpn (X) = f(�)
n
r+l! 8,, (x - t) dt = f(�) . n
But f is uniformly continuous on the compact interval K. So, for every c > 0 there exists an integer r such that I f (x) -f (x ) 1 < £ , whenever '
1
x, x ' E K and I x -x ' I < - . Hence I f(x) - ({Jn (x) I = l f(x) -f(;) I < £ for r all x E J and n > r, which proves the lemma
.
•
Let I be any interval, p an integer � 0 and f E C P(I) . Then there exists a sequence of functions ({Jn E C oo (/R) s uch that ((I (k) � f (k) uniformly on each compact interval J C l , for n k = O, . , p. 6.8.3. THEOREM.
. .
1 22
PROOF.
a) Suppose
I = IR. Consider the sequence
in the lemma. Observing that
�l defined as
DX �I (X- t) =-Dt� (x - t) and �;k) (X_ t) = 0 for I x - t I > � , k = O, 1 , . . . , n
it is easily seen that ( k) ((In (x) =
f On(k) (x - t) f (t ) dt JIR
= (- l l
=
JfIR [Dt(k) �1 (X - t)] f(t) dt
f �(x - t) f (k)(t ) dt , JIR
Vx E IR , k =O, 1 , . . . , p, n = l , 2, . . . .
Now, applying the lemma to the functions f (k), it is readily seen that (k) k ((In � f ( ) uniformly on each compact interval. b) S uppose I is closed. Then it is possible to extend f to a
JE CP(lR)
I is bounded, it is possible to make f equal to two polynomials outside I). Now, if we set
function
{fin
=
f JIR � (x
-
(for example, if
t) J( t) dt and ({I = PI (fi the theorem is proved in this n n
case. c) Suppose ping
I is
h of IR onto I.
open. Then, there exists a one-to-one So if we put g = f 0 h , f!In
=
({In = f!In 0 h- 1 the theorem is proved in this case.
fJIR � (x
-
C map<Xl
t ) g (t) dt and
I is half-open. Then it is possible to extend f as a continuous function f on an open interval I -:J I, which red) Suppose finally that
duces the proof to the previous case. •
,....,
1 23
We are going to establish a similar theorem for the space C; (I) but, for that purpose, it is convenient to prove, first, a lemma. It is sufficient to consider the case when I is compact, 1 = [a, b] .
6.8.4. LEMMA. Let f be a C oo function on I such that fCk)(a)=fCk)(b)= O for k 0, . . . , p . Then there exists a sequence offunctions CP E C; (I) such
that CPnCk) ďż˝ f Ck) uniformly on I, for k = O, . . . , p .
=
PROOF.
4
b-a
and I
,
is given by 6 . 8 . 1 . It is easily seen that for
we have
an (x) I
an(X)=Hn(x-a- :) - Hn (X - b + :) \:/x E IR , Hn [a + -2 , b - -2 ] an (x) =0 I, an (x) = n n ,
Set
n = 1 , 2, . . . , where n>
n
outside
s i for all
x.
1 on
On the other hand, since
.
Hn (x) =y (nx),
[ , b- -;;] 2
have H Ck) (x)=n ky (k) (nx) for k= O, . . . , n. Then lf we put In= a + n -;; and M
n>
4
-
b a
max ( l y(x) I , . . . , l yc P) (x) I ),
- l :s x s l
we
2
it is readily seen that for all
and k= O, I , . , p ,
..
6 .S.S.
Thus, set CP =
In for n > 6 .8. 6 .
. b-a 4
n
anf
n E C; and CPn = f on
for n = 1 , 2 , . . . . Then CP
On the other hand:
1P:'J - f ('J =
i: (k)(an lr C k -V\ f v
v=o
.
for k = O, 1 , . "
But since f Ck)(a) = f Ck)(b) = 0 for k = O, 1 , . " p , it follows that:
.
1 24
f (k) (X) . f (k)(X) lzm - = 0 for k=O, l , . . . , p - = lzm p (x - a) P k (x - b) k .
x -+ b -
x -. a +
and therefore there exists a sequence of numbers and:
I f (k) (X) I S
��k n
on I\In ' for
en such that en � 0
k= O, . . . , p .
From here, from 6 . 8 .5 . and from 6 . 8 . 6 . , follows :
I lI'n(k) - f (k) I s 2k ( 2M + 1)
��k '
n
for
k= O, . . . , p
.
•
For every fE C� (I), there exists a sequence of functions lI'n E C; (I) such that lI'n converges to f in the norm 11 · 11 p. 6.8.7. THEOREM.
fE C� (I) . By 6. 8 . 3 . , there exists a sequence of functions If/" E C 00(1) such that If/" � f in the normed space CP(I). PROOF. Consider
We have also seen in the proof of 6.7.9. that there exists a continuous projection 7r of C P (I) onto C� (I) such that 7r(qJ) qJ is a polynomial for every lI' E C P (I). Set Xn= 7rIf/,, ; then Xn E C OO(I) for all n and II Xn- f Il P� o. Finally, by the lemma, there exists for every n a se quence of functions Xn 1 Xn , , belonging to C; (I) and converging 2 to Xn in II · II P . Then, from the double sequence Xn , we can select a se k quence of functions lI'n E C; (I) converging to f in the norm li P . • -
'
•
•
•
1·
Consider now a distribution f on IR and set:
lI'n (x) =
f 8 <x- t) f(t) dt JIR n n
where 8n is given by 6 . 8 . 1 . Then if f = D F with FE C(IR) , it is read il y seen that:
and from 6. 8 .2., it is concluded that qJn � f in the distributional sense. This result can be extended to every f E PJ (IR) observing that on
1 25
every compact interval I, f reduces to a distribution. Finally, if I is any interval in IR , we can see by the technique used in the proof of
6.8 . 3 . , that:
6. 8.8 . THEOREM.
For every f E !if (I) there exists a sequence of functions fPn E C O) (I) such that fPn � f. Remember that the space !if (I) is complete. Theorem 6 . 8 . 8 . has suggested to Mikusinski a construction of the space !if (I) by com pletion of C O) (I) with respect to the distributional topology. Accord ing to this approach, a fundamental sequence is a sequence of func tio ns fPn E C O) (I) such that, for every compact interval J C I, there of functions lPn E C O) (I) (depen ex ists an integer p and a sequence n dent up on J) such that fPn= D lPn on J and lPn is uniformly conver gent on l. Two fundamental sequences ( fPn ) and ( fYn ) are said to be equivalent if and only if for every compact interval J C I, there exists an integer p and two sequences of function lPn ' fYn in C O) (I) such that fPn DP lPn ' fYn DP � and lPn � � 0 uniformly on J. This turns out to be actually an equivalence relation. Hence the corresponding equivalence classes are called distributions (i . e . global distributions according to our terminology). =
=
-
REFERENC E S
' [ 1 ] S . LOJASIEWICZ. Sur la va leur et la limite d une distribution en un point. Studia Math. 1 6 ( 1 957) pag. 1 -36. [2] 1. MIKUSINSKI-R. SIKORSKI . The Elementary Theory of Dis tributions. Panstrowe Wydunictwo Nankowe, Varsaw, I ( 1 957), 11 ( 1 96 1 ) .
[3] 1. S EB ASTIA o E SILVA.
la theorie des distributions.
Sur une construction axiomatique de Revista da Faculdade de Ciencias de
Lisboa ( 1 954-55) .
[4 ] L . SCHWARTZ.
Theorie des distributions, I, ll. PARIS ( 1 950-5 1 ).
1 26
[5] S . L. SOB OLEV. Methode nouvelle Cl resoudre le probleme de
Cauchy pour les equations lineaires hyperboliques normales.
Mat. Sbornik 1 (43), 3 9-72, ( 1 936) .