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.
C H A PT E R I X
FO U R I E R TRAN S FO R M ATI O N .
9 . 1 . Fourier transformation for tempered distributions on fR.
Let
f be any distribution on fR. If the integral
convergent on
r eiXYf( y) dy is J/R
IR , then the distribution
9. 1 . 1 .
g (x)
=
r e ixY f( y) dy JIR
is called the Fourier transform of f and we write g (x ) =
simply g
�� I Y f( y) , or
�f. Frequently the Fourier transform of f is also denoted by f . For simplicity we shall omit the subscript IR in the integral sign "
=
when no confusion can arise. As an example, we have seen that tional sense. Hence,
J
e ixy 1 y2 dy ne - l x l in distribu+
=
1 80
From this we deduce that
feiXYdy
=
21Ct5(x),
and so
�1 = 21!o.
9.1 .2.
On the other hand, it is readily seen that 9.1.3.
We now establish some fundamental properties of the Fourier transform. 9. 1.4. lf �f and �g exist, then � ( A. f + J1g ) exists for all A. , J1 E C and �( A. f + J1g ) = A. ( �f ) "+ J1 (�g ) . PROOF. This is an immediate consequence of the linearity pro
perty for integrals . •
9.1.5. lf �f exists, then c.J (Df ) exists and � (Df ) = - ix (�f ) . PROOF. e i XY f ' ( y) = D ( eiXY f ( y» ixei xY f( y). If �f exists , i.e. , if y eixY f( y) is integrable on IR, then e ixY f ( y) -.:, 0 on IR as y -.:, 00 , and -
therefore
feiXY!'( y) dy = - ix fe x i
Y f ( y) dy . •
9.1.6. lf �f exists, then � ( Yf ) exists and � ( Yf ) = - i D (�f ) . PROOF. If :ff exists, then by the differentiation property
f
- i Dx e ixy f ( y) dy Combining
9. 1 .4., 9. 1 .5.
and
=
fe X y ( y) dy . i
9. 1 .6.,
y f
gives :
•
9. 1.7.
181
If P is any polynomial, then
CS (P(D)f ) = P(- ix) (CSf ) CS(P( y)f ) = P(- iD) (CSf ) . We now establish some existence criteria for Fourier transforms . 9.1.8.
If f is summable on /R, then CSf exists and is a bounded
continuous function.
PROOF. Suppose
x, y E IR ,
Y
Y
Since
f i eiXYf( y) i dy
the integral
f i f( ) i d feixYf( y) dy
f E L(lR).
l e iXYf( y) I = l f( y) 1
for all
is dominated by the integral
which is convergent and independent of
x.
Hence,
is uniformly convergent on IR , and thus it is convergent
in the distributional sense and represents a continuous function
on IR . Finally,
i g (x) i s i f( Y) i dY for all x EIR .
f
We shall denote by
g (x)
•
Cb the space of all bounded continuous func'-"
tions on /R. Recall that § denotes the space of all tempered distributions on 9.1.9.
IR . From 9. 1 .7 . and 9. 1 . 8 . follows : '-"
'-"
If f E§ then CSf exists and CSf E§. '-"
PROOF. Suppose f E§ . There are
such that
f = D mF
Set lP =
F ( 1 + ix ) p + 2
and .
F E O(xP )
Then
m, pE/No
and
in the ordinary sense
FE C(lR) as x -;. 00 .
f = D m« 1 + ix )p + 2 lP) , lP E C(/R),
and
1 82
Therefore, by 9. 1 . 8 . , �f l/J exists and J l/J E Cb C [:g . Hence, by '--'
9. 1 .7 . , �ff also exists and :ff (- ix)'1!( 1 + D) p + 2 (Jl/J) E !Z1 . '--'
=
•
We next propose to study the problems of the inversion of ;t. We observe that :f transformed 1 into 2 no, 0 into 1 , D into mul tiplication by - ix, and multiplication by x into - iD. Hence, if exists, it must transform 0 into 1 /2n., 1 into 0, etc . Thus we might expect that � - l is given by the formula:
:-f - 1
9. 1. 10.
f ( y) = 1 2n
f.
e-1XYg (x) dx .
We shall temporarily denote by :f * the transformation g ---+ f de fined by 9 . 1 . 1 0. It is readily seen that � * has the required properties and that 9.1. 1 1 .
� * f exists for all f E �. Moreover, '-"
'--'
'-"
If f E !?L? and g = ;}j, then j = ;} *g ; conversely, if g E � and
j = ;} *g , then g = Jf. PROOF. Suppose j E !2J and set g = :fj, h = �f *g . Then '--'
and, if we may interchange the order of summation, we find
h ( y) =
1 2 n:
f(f
)
e 'x ( Y'-Yl dx f ( y ' ) dy ' .
But
f
e 'x ( y'-yl dx = 2 n:8 ( y ' -y) = 2 n:8 ( y - y ' )
and by Dirac ' s formula
h ( y) =
f
8 ( y - y ' ) f( y ' ) dy ' = f( y) .
It is shown analogously that if
1 83
and f = J*g , then g = Jf. We need only justify the interchange of summations, and by 8.4.4. it is sufficient to show the convergence of the double integral '-'
g E fZJ·
ffeix( Y'-Y)f( y' ) dx dy'.
9 . 1 . 12.
The integral 9. 1. 13.
with
f E L , is uniformly convergent on fR since for all x, y, y ' E fR , _
1 1 + X2
and
l e iXY 'f( y ' ) 1 = 1 f( y' ) 1 ,
and the functions ( 1 + X 2 )- 1 and f ( y' ) are summable on fR . Hence, by applying the operator 1 - D 2 to 9. 1 . 1 3. , we see that 9. 1 . 12. is con'-' vergent for f E L . The result for f E !if now follows by an argument similar to the proof of 9. 1 .9. if we observe that 9. 1 . 10. represents '-" :f* J and that J * J (Df ) = Df, J * J (x f ) = x f , for all f E .§?) . •
Thus we have proved that J * = J- 1 for J restricted to !if. We ask if tempered distributions are the only distributions having a Fourier transform in the previous sense. The answer is affirmative: '-"
9. 1 . 14.
If the
integral e i XYf ( y) dy is convergent on fR , then f E jJj .
PROOF. Suppose
f feiXYf( y) dy
is convergent on fR . Then
is of the form ( 1 + iy)- I D mD n F(x, y) where F(x, X y on each bounded interval as y � 00 . Hence,
eiXYf( y)
y) E O(yll) uniformly
1 84
and, since the right member is independent of f E O(y2 n -l ) and so f E .@" . • '-"
x,
it follows that
The preceding results may be summarized as follows :
J is a one-to-one linear mapping of the space .@" onto itself, changing D into multiplication by - i x , multiplication by x into -iD, 1 into 21CO, and O into 1 . J-1 is given by 9 . 1 . 1 0 . 9. 1 . 15. THEOREM. '-"
9.2. Fourier transformation and convolution.
The following theorem is well-known:
If f and g are summable functions on IR , then J transforms the convolution f * g into the usual product of the continuous functions Jf and Jg. That is, 9.2. 1 . THEOREM.
By the theorem of Fubini-Tonelli (8.7. 3 .): if f , g E L, then f * g exists and f * g E L . Let j = Jf, g = Jg. Then by 9. 1 . 8 . , f , g E Cb and PROOF.
"
"
j (x) g (x) = r e iXUf(u) du r eiX Vg(v) dv =
Jm
Jm
r e ix ( u + v )f(u)g( v) du dv. Jm2
Now let u + v = y, v = t. Then u = y - t, the lacobian of the transformation is 1 and the transformation maps IR 2 onto IR 2 . There fore,
j (x) g (x) and so
=
(
)
r eixYf( y - t) g(t) dydt = r e iXY r f( y - t) g(t) dt dY , Jm2 Jm Jm
jg = J( f * g) .
•
1 85
9.2.2. COROLLARY .
Let !, g be distributions on IR of the form f = DmF, g = D n G, where F and G are locally summable functions satisfying the condition that there exists an integer p such that ( 1 + ix ) P F and ( 1 + ix )-P G are summable on IR . Then J( ! * g) = = ( J! ) (Jg). This is a consequence o f theorem 9. 2 . 1 . and properties 8 . 5 . 8 . and 8 . 5 . 1 0. The corollary can obviously be extended to distributions which can be expressed as finite sums of the preceding forms . Recalling the definition of the space fiJ of all rapidly decreasing distributions, it is easily deduced from 9 . 2 . 2 . : .
-..
9.2. 3. COROLLARY.
-..
'-'
If ! E fiJ, g E fiJ, then J ( ! * g) = (J! ) (Jg).
In order to characterize the Fourier transforms of the rapidly de creasing distributions, we shall first establish two general criteria: 9 .2.4. THEOREM.
If ! is a distribution of the form D nF, where F is a locally summable function on IR and FE O(�-r) for r, an integer � 2, and if (jJ= J!, then (jJ is a C r - 2 function and (jJ (k) E O(x n ) for k = 0, 1 , . . . , r- 2 . PROOF. Suppose the hypothesis is satisfied and put qJ= JF. Then (jJ= (- ix )nqJ and since xkFE O(x - 2 ) for k = O 1 , . , r - 2, it follows that D k qJ E Cb for k = 0, 1 , . . . , r- 2 by 9. 1 .6. and 9 . 1 . 8 . Hence (jJ E C r - 2 and (jJ (k) E O(x n ) for k = O, 1 , . . , r-2 . • ,
. .
.
9 .2.5. THEOREM.
If (jJ is a C r function such that (jJ (r) E O(x n - r ) for n, r EINo , and if! = J(jJ, then ! is of the form ! = (1 + D) n + 2Ffor FE C
such that FE O(X-r).
PROOF. S uppo se the hypothesis is satisfied and put
QJ= ( 1 + ix )-n-2(jJ, F= JqJ. Then ! = ( 1 + D)n + 2F. On the other hand, (jJ E O(x n - k ) for k = O, 1 , . . . , r , and this implies (jJ (r) E o(x - 2 ). Hence x rFE Cb and so FE O(X-r) . • 9.2.6. DEFINITION. A tempered
C OO function on IR
is a function
1 86
ifJ E C r¥J (lR) satisfying the condition that for every r = O , 1 , . . . , there exists an integer n such that ifJ (r) E O (xn ) in the ordinary sense as x� We denote by iVl the set of all tempered C r¥J functions on fR. 00 .
It is easily seen that iVl is a vector subspace of � n C serve that iVl �§?J n C From 9.2.4. and 9. 2 . 5 . we have: '-'
'-'
GO
;
but ob-
GO .
The Fourier transformation J maps the convolution algebra §?J onto the multiplication algebra iVl . -... PROOF. a) Suppose f E 2J". This implies that for every r = O, 1 , 2 ,
-...
9.2.7. COROLLARY.
. . . , f can be represented in the form f =
k D" F. m
where
F. E O(x )
- ,- 2
for k = 1 , 2 , . . . , m. Then if ifJ = ;Sf, it is easily seen from 9.2.4. that ifJ E c r and ifJ (r) E O (x JJ ) where J1 = max (r J , . . . , rk , . . . , t:n )' Hence,
ifJ E iVl .
b) Suppose ifJ E iVl . Then for every r = O, 1 , 2 , . . . , there exists n such that ifJ(r) E o(x n ) . Thus if we put f = ;S * ifJ, we conclude from 9 . 2 . 5 . (which obviously extends to ;S * ), that f is of the form ( 1 + D)n+ 2F, -... where F is a continuous function such that F E O (x-r). Hence, f E §?J. •
9.3. The Fourier transformation as a continuous mapping.
It can be seen that the Fourier transformation is not continuous with respect to the topology of § restricted to 12J. However, we can define a stronger topology on §?J which will make ;S as well as D continuous, and extends the usual topologies on function subspaces of 12J. In the space Cb of all bounded continuous functions on fR a norm is usually defined by I l f ll = sup I f(x) l . Then, '-"
'-'
'-'
x E IR
9.3 . 1 . LEMMA. ;S
space
L
defines a continuous mapping of the normed into the normed space Cb '
1 87
PROOF. It is sufficient to observe that if f E L , then
II 3'f II s l f l = I l f I l L (cf. proof o f 9 . 1 . 8 . ) .
f
• "-""
We shall try to define the strongest topology on !?lJ making both � and D continuous and inducing a topology on Cb (resp. L) weaker th an the norm topology of Cb (resp. L) . If such a topology exists, then �-l and the mapping f � x f will also be continuous . These considerations lead us to the following definition of con vergence for sequences : "-""
A sequence of distributions { fn } c!?lJ converges in the tempered sense to a distribution gE !?lJ if there exist an integer p, a s equence of functions { Fn } C Cb and a function G E Cb such that (i) fn= DPFn for all n ; (ii) g = DPG ; (iii) ( 1 + x 2 )-P(Fn- G ) converges to 0 uniformly on IR as n � oo.
9.3.2. DEFINITION.
"-""
It i s now a simple exercise to verify that this concept o f conver gence satisfies all of the preceding conditions . In order to define in !?lJ the strongest topology satisfying the same condition, we shall denote by cb-r for k = O, 1 , 2, . . , the space of all distributions of the form f = Dr( 1 + X 2 YF with F E Cb ' and we shall consider cb-r provided with the image topology of Cb by means of the mapping F� D r ( 1 + x 2 YF of Cb onto Cb-r. Then it is eas ily seen (as in the case of distributions on a compact interval) that cb-r is a normed space and the inj ection Cb-r� cb-r-l is compact for r = 0, 1 , 2, . . . . On "-""
.
the other hand 15
00
U Cb-r so that jffi- with the inductive limit topology
r=O
of the normed spaces cb-r is an (LN*)-space. Then it can be seen that this topology is the strongest one satisfying all preceding conditions and such that the concept of convergence for sequences agrees with that defined directly in 9.3.2.
188
It can also be proved that the substitution
x=t/(t2-1) defines --a
one-to-one continuous linear mapping of the locally convex space ďż˝ into the locally convex space.2? [-1,
1].
9.4. Fourier transformation and scalar product. Sometimes the Fourier transformation is defined by the formula
9.4.1.
g(x)=
1
v'2n-
L.
e,xYf(y)dy,
IR
instead of 9.1.1. So far as no misunderstanding may arise, we shall still write in this case g = 'Sf . The fundamental properties of Fourier transformations that we have previously proved are not altered by this change of form. But we now have of course,
The advantage of this new form is that it preserves in many cases the hermitic scalar product of two distributions.
9.4.2. DEFINITION. A rapidly decreasing function
CO'J
l/JECO'J such that l/J(n)EO(x-r) for all n,
function on
r=O,
IR
1, 2, ... .
is a
We shall denote by S the set of all rapidly decreasing functions.
--
S is a proper vector subspace of.2? n 0lL. For example,
It is easily seen that
(f, l/J) exists
exp(-x ) ES. on IR whenever fE!!if and ifJ ES.
Moreover, if we consider the topology on
S defined by
2
'--'
the sequence
of norms
IlifJll = sup ( 1 l/J(x)l, (1 +x2)lifJ'(x)I , . . . , (1 +x2)nlifJ(n)(x)I), n xE1R
it can be proved that !!if is isomorphic to §'. (In the theory of Schwartz, '-'
189
the space ffJ is defined to be '-"
Sf).
On the other hand, it is easily seen
by applying 9.2.4. and 9.2.5., that Thus
j=
maps the space
S
onto itself.
S is at the same time a multiplication algebra and a convolution
algebra.
9.4.3. THEOREM. PROOF. Set
(f, ifJ)= =
If fE� and ifJ E S , then (f , ifJ)= (j=f, j=ifJ).
g=j=f, If/=j=ifJ. Then,
LIR f(X)ifJ(X)dx=LIR (� JIRf e-iXyg(Y)dY) ifJ(X)dX 21!
L
1
)
f e-ixYg(y) ifJ(x)d xdy= f g (y) ( 1 e-iXYifJ( X)dX dY � JIR2 JIR \� IR
= f g(y) If/(y)dy,
JIR
since the double integral exists and
exp(-ixy) = exp(ixy) .•
In the theory of Schwartz, this theorem is true by definition since
� is defined as the transpose of j= restricted to
(j= f, ifJ)= (f , j=ifJ),
for all
S:
fE�, ifJ ES.
Let us now consider the Hilbert space of all square summable functions on
IR,
which we shall denote by j(. We shall put
11 f 112= v(f, f )
for all
fE j(.
Convergence in this norm is called convergence in the square
mean. It is well known that every
fE j(
square mean by a sequence of functions
can be approached in the
{ifJn} CC*OO(lR),
so that in par
S is dense in j-c. On the other hand, it follows from 9.4.3. that if ifJ ES, then Il ifJ112= IIj=ifJI 1 2' Consequentely, the Fourier transforma tion restricted to S can be extended to a linear isometry of the space ticular
1 90
J( onto itself. We shall provisionally denote this "mapping by � . In particular, if f is a locally summable function with bounded carrier, then clearly fE J( and �f = �f . Thus, in general, ?If is given by the + limit, as a - and b in the square mean of �
00
�
00 ,
1
21! I
v'21Z 9.4.4. LEMMA.
be f(y) d iXY
a
y
.
L:f(x-y)g(y) dy converges uni-
If f, g E X then
formly on each compact subset of fR , as n � + 00, to a continuous function h such that � h = (�f ) (�g).
fn(x) = f(x)(H(x + 2n) - (H(x - 2n» ? gn(x) = g(x) (H(x + n) - H(x - n). Then fn ' gnEL n J( for all n and PROOF. Set
L:f(x -y)g(y) dY = ( fn* g. ) (x) ,..,
--
-...,
for I x l < n. Hence, if we put "
"-'
in = Jfn '
I'\.
gn = �g n ' f = � f and g = � f , we have fn g n= � ( fn* gn ) EL for all n, and since fn � f , g n g in the square mean, then fn g n � f g in the square mean, and therefore fn * gn h = � - 1 (f g ) uniformly on fR . "'"
___
A
�
"""-
�
Consequently,
L> (X -y) g(Y) dY
converges uniformly on each
COffi-
pact subset of fR to the function h, which is obviously continuous . • 9.4.5. THEOREM. Every function
f E J( is a tempered distribution
and �f = �f for every f E J( . Moreover, convergence in the square mean implies convergence in the distributional sense and if f , g E J(, then f * g exists in the distributional sense and is a continuous func tion such that � (f * g) = ( � f ) ( � g) .
191 -
",-,
"""'"
A
--
-
PROOF. Set f = J f , fo = ( 1 + iX )-,l f and fO = J -l f o ' (observe that
fo E 3CnL).
Since J-1 ( 1 + ix ) = O- O' , ( O- O' ) *fo = ( l - D)fo ' it fol--
lows from the lemma that f= ( l - D ) fo and hence f E §Z1 , since fo E Cb .
The remainder of the theorem follows from the preceding results . 9. 4.6. COROLLARY. If f,
g E J-( , then
( f,
g ) = ( J f, J g ) .
PROOF. It is sufficient to observe that J is an isometric linear m apping of the hermitic space J-( onto itself. • 9. 5. Fourier transformations on IRn.
The Fourier transformation on /R n may be defined by
where f is a distribution on IR n, and xy = convergent on /R n , we write
� xk n
Yk "
If the integral is
g = Jf.
A distribution f on /R n is said to be tempered if and only
if there exist two systems p, r E /Non and a function
that f = DPF and --
F E O(x(' . . . x;n ) --
F E C(lR)
such
in the ordinary sense . We write
fE §Z1 (1R n ) or simply f E �.
All preceding properties of the Fourier transformation can be extended to the present case with the obvious modifications concern
ing the existence of n derivation operators and n coordinate functions "
XI "
'"
"
x ' Thus, n
J (Dk f ) = (- i xk) ( J f ), J (xk f ) = (- i Dk )( J f),
1 92
--
for all fE � and
k= 1 , . . .
, n.
Moreover, in the inversion formula the
1
1
. coefficient - must be replaced by ( 2 1f)n 2 1f
--
Observe that if fE @(IR n ) we can define
the Fourier transform of f with respect to Xk • (It is easily proved that this partial integral is convergent on
J]
J�
and it is easily seen that
et = 'J1 . . . Ietn
J
IRx) · Then we write gk = 'Jkf,
•
--
For the existence of 'Jkf it is not necessary that f E @ (lR n ) . It is sufficient that there exist an integer p, such that
! E O(Xf ) on
J]
J�
IRx as j
Xk � OO .
1 93 SUPPLEMENTARY BIBLIOGRAPHY
[ 1 ] A.
Sur une fafon de definir sans dualite l ' espace des distributions temperees sur la droite et la transformation de Fourier. Portugalire Mathe m atic a 1 8 ( 1 959), 1 25- 1 53 . ANDRADE GUIMARA.ES .
,
[ 2 ] J . SEBASTIA. O E SILVA. Les fonctions analytiques comme ultra-distributions dans le calcul operationnel. Math . Annalen, vol . 1 36, pp 58-96, ( 1 958).
[3] J. SEBASTIA.O E S ILVA . Sur le calcul symbolique des operateurs differentiels a coefficients variables. Rend. Accad. Lincei (8), 27 ( 1 959), 42, 1 1 9- 1 22 .
[4] J.
SEBASTI A. O E SILVA . Sur " la definition et la structure des distributions vectorielles. Porth . Math . , 1 9 ( 1 960), 1 -80.
permutables a spectre vide ou non-borne. Annali di Matematica Pura ed Applicata, (4 ) , (58) ( 1 962), 2 1 9-275 .
[5] J . SEBASTIA.O E SILVA . Sur le calcul symbolique des operateurs
[6] I .
MARINESCU. Espaces vectoriels pseudo-topologiques et theorie des distributions. Deutscher Verlag der Wissenschaften, Berlin, 1 963 .