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English Pages 219 [231] Year 2016
A n n a l s o f M a t h e m a t i c s S tudies
Number 19
ANNALS O F MATHEMATICS STUDIES Edited by Emil Artin and Marston Morse 3. Consistency of the Continuum Hypothesis, by KURTG ~ ~ D E L 7. Finite Dimensional Vector Spaces, by PAULR. HALMOS 11. Introduction to Nonlinear Mechanics, by N. KRYLOFF and N. BOGOLIUBOFF 14. Lectures on Differential Equations, by SOLOMONLEFSCHETZ 15. Topological Methods in the Theory of Functions of a Complex Variable,
by MARSTONMORSE
16. Transcendental Numbers, by CARL LUDWIGSIEGEL 17. Problkme GCnCral de la StabilitC du Mouvement, by M. A. LIAPOUNOFF 18. A Unified Theory of Special Functions, by C. A. TRUESDELL 19. Fourier Transforms, by S. BOCHNERand K. CHANDRASEKHARAN 20. Contributions to the Theory of Nonlinear Oscillations, edited by
S. LEFSCHETZ
21. Functional Operators, Vol. I, by JOHN VON NEUMANN 22. Functional Operators, Vol. 11, by JOHNVON NEUMANN 23. Existence Theorems in Partial Differential Equations, by DOROTHYL.
BERNSTEIN
24. Contributions to the Theory of Games, edited by H. W. KUHNand A. W.
TUCKER
25. Contributions to Fourier Analysis, by A. ZYCMUND, W . THANSUE, M. MORSE,
A. P. CALDERON, and S. BOCHNER
26. A Theory of Cross-Spaces, by ROBERTSCHATTEN 27. Isoperimetric Inequalities in Mathematical Physics, by G. POLYAand
G. SZEGO
FOURIER TRANSFORMS BY S. BOCHNER AND K. CHANDRASEKHARAN
PRINCETON PRINCETON UNIVERSITY PRESS LONDON: GEOFFREY CUMBERLEGE OXFORD UNIVERSITY PRESS
1949
Copyright 1949 Princeton University Press
Photo-Lithoprint Reproduction NEW YORK LITHOGRAPHING CORP. NEW YORK, N.Y.
PREFACE T h i s i s a t r a c t d e a l i n g w i t h F o u r i e r t r a n s f o r m s and some t o p i c s n a t u r a l l y co n n e cte d w i t h them, and a lt h o u g h the m a t e r ia l included i s f a m i l i a r ,
i f not c l a s s i c a l ,
th ere
i s not much o f a d u p l i c a t i o n w i t h o t h e r books i n t h e f i e l d . Acknowledgement o f th a n k s i s due from B o ch ner t o t h e O f f i c e o f Naval R e s e a r c h , and from C h a n d ra s ek h ara n t o th e I n s t i t u t e f o r Advanced S t u d y .
P rin ceton U n iv e r s ity and The I n s t i t u t e f o r Advanced S t u d y . November 19^8.
ERRATA page 1
6
f ( )
2
k
tR —00
13
2
ga"£
5
16
1
1 8
10
21
read
fo r
lin e
4>f (oO
SR
2
IH( t )
lH(t) |
isg (o )
Is r ( ° )!
^hR 21
39
57
11 — CO
—oo
12
-(i x f (o' (ex).
i f i G x ) = -xcxcj) (cx); a l s o oo (U.1 )
f ( x ) = -J
f'( x ) d x X
P ro o f:
( i ) We have ix h d)(ex +h) - J)(oc) = T [ f (x ) . e - ■ ~1 ] h 11 = T [ f h (x ) ],
say.
Now, f ^ ( x ) ~7>ixf (x ) i n L ] -norm, b e c a u s e f ^ ( x ) - ^ i x f (x ) a t e v e r y p o i n t x , and I f h (x )|
ixh < I f ( x ) | l g— -^
1
i
|x|
. |f(x )| € L1 .
On a p p l y i n g p r o p e r t y ( 1 . 5 ) we g e t T [ fh (x)] u n i f o r m l y , a s h-^0.
— > T [ i x f (x ) ],
Hence, a t e v e r y p o i n t ex , t h e r e e x i s t s
t h e d e r i v a t i v e d)! (cx) i n t h e o r d i n a r y s e n s e , and
§4 .
D e r i v a t i v e o f a f u n c t i o n and i t s t r a n s f o r m
9
ix f («) = ' (oc) . (ii)
The p r e c i s e meaning o f o u r a ss u m p tio n s i s t h a t
t h e r e e x i s t s a f u n c t i o n g ( x ) £ L 1 w h ich we ch o o se t o d e n o t e b y f * ( x ) and an i n d e f i n i t e i n t e g r a l o f i t f(x)
= $ g(y)dy
such t h a t f (x ) €, L 1 ( -00 ,00 ). Now, A f(A) - f ( a ) = J g(x)dx a I f we k eep a f i x e d , and l e t A-^oo , s i n c e g ( x ) £ L 1 , we have A $ g(x )dx — > c. a T h e r e f o r e , f ( A ) - ^ l ; s i m i l a r l y f ( - A ) - ^ -m, s a y .
S in ce
f ( x ) € L 1 (-0 0 ,0 0 ) we must h a ve 1 = -m = 0, and t h i s , f i r s t of a ll,
proves
( 4.1 ).
N o w , i f T [ f * ( x ) ] = y((cc) =0
fo r
a lm o st e v e r y w h e r e .
p ( x ) be t h e f u n c t i o n d e f i n e d a s f o l l o w s :
1,
-a ^ x £ a ;
o, x > a+e., x < - a - t ;
12
I.FOURIER TRANSFORMS IN L
(ONE VARIABLE)
Let
T[fW
( x ) ] = Y a , £ (o^
so t h a t
00
fa ,e ^
= 2 $ ga , ^ x ^ c o s xoc d x 00 r " (oc)doc s i^ R t d t
SR( x )
i n t h e n o t a t i o n o f §5(7 . 8 )
H (t)
1
1
0
0
2 f ' ( l - a ) c o s o c t doc = 2 f (1
2_ 1 t
-COS t
t
-oc)d
16
I.
FOURIER TRANSFORMS IN L S (x) = — - J ( 1 R 27r -R =
R S
=
R f d - f)d
J 0
i R ^ f S 0
=
( 7 -9 )
- I I K((cx)doc R .,
H
H (t) =
(ONE VARIABLE)
«
v- ioo, and i f O 1 oo f u r t h e r m o r e (8 ) 1 = ^ To JC H ( t ) d t ; t h e n , a t a p o i n t x , t h e -oo c o n d itio n
i
^
j
h
g x ( t ) d t = 0(1 ), as. h -> o
im p lies SR( x ) N ote t h a t H ( t ) a s w i l l be proved
is
- f (x ) = o (1 ) , a s R - ) oo . even,
s i n c e K(oc) i s
even,
and t h a t ,
s u b s e q u e n t l y i n th e o r e m 9, a s s u m p t i o n ( 8 )
is a consequence o f th e p re v io u s
ones.
§7 P ro o f:
Su m m a b ility theorem s
From ( 7 . 6 ) we have
1
K S (x) R
oo
- f(x ) = ^ f o
g ( t )RH(Rt )dt x oo
o
♦
S
1
+ I 2, say.
u S
=
= I
u
I f we put
5
G (t) =
g (s)d s,
we have I,=
$
RH(Rt)dG
= uR . H(Ru)
^ G ( t ) R dt H ( R t ) o
= o(i ) +
=
0(1)
+
0(f
u tR d .H (Rt )) o
0 ( uR
u . H(Ru) - J R H (Rt)dt)
and f i n a l l y (
7-11
I t = 0(1 ),
)
by a ss u m p tio n s T
2
=
1
( 6 ) and ( 7 ) .
A lso
r°° ^f ( x + t ) + f ( x - t ) _ g f ( x ) ]RH(Rt)dt
71 u
2
= I 2 ,1 + I 2 ,2 + * 2 ,3 ’ s a y ' II2 ,l
{
+° (15.3 )
1
f* r\ "P F or e v e r y y > 0, t - ,— ^ e x i s t and b e l o n g t o L 1 6x
F or i n s t a n c e ,
and f o r f i x e d y )> 0 t h i s
is
00
^
f(z)h (x -z )d z
-00
where h ( x ) € L 1 and hence 5“ £ L^ by theorem 2; s i m i l a r l y fo r su ccessive d e riv a tiv e s
§1 5 *
Boundary v a l u e s
b]
6(r)f 6x2 o f any i n t e g r a l o r d e r r . ( 1 5 -M
V JT»
F o r e v e r y y > o, t h e r e e x i s t s a f u n c t i o n
C L1
such t h a t Um h — >0
f001 -oo
. ££ | dx = h^
o.
We mean t h a t t h e l i t e r a l p a r t i a l d e r i v a t i v e o f f ( x , y ) w i t h r e s p e c t t o y i s t h e l i m i t i n norm o f t h e d i f f e r e n c e q u o tien t.
-
For,
_( * - z ) 2
00
v t
L
f(z)y
2 e
^
dz
. 00 77 = \ f (x+z )(y,z )d z , -oo
=
where
Hence r.
| f (x ,y+h ) - f (x , y ) _ 6 f
-oo
(y+h,s)-(|>(y,Z ) _ h ,
|| f || . 5°° | ^ ( y + h .z ) - j i ( y , z ) _
| dz
Now we know t h a t l lm h
i ( y + h , z )-j)(y , z ) = o
h
. 6y
b i
^
,
h2
I.
FOURIER TRANSFORMS IN L1
to
show
th a t th is
to
show
th at,
(ONE VARIABLE)
r e l a t i o n h o l d s i n L 1 -norm ,
fo r
fix ed
y >
I
it
It
( 15. M )
su ffice s
to
enough
0, ^ I < 7 (z) e
u n ifo rm ly in h.
is
l,
show s e p a r a t e l y t h a t
I 5^ I < y ( z ) C L 1
and t h a t
(1 5 . 4 2 )
|
I
^( y , z )
where th at
=
, and t h u s
( 1 5. M ) h o ld s u n ifo rm ly
v e rifie d (15.5)
^ (J) (y+ t ,2 )dt O
( 1 5 . ^2 ) f o l l o w s from th e f a c t in
(1
) > a s c a n a g a i n be
fr o m ( i 5 *)+iO For e v e ry y >
0,
V2
Lf
— -= bx
t-y
CD
(15 .6 )
5•
C -00
(by d i r e c t
oo
!f ( x , y ) |dx < M (= J -00
if(x )|d x )
com p u tatio n ).
§1 5 (b y Lemma
Boundary v a lu e s
k
3
2 ).
THEOREM 2 3 . F o r g iv e n f ( x ) £ "L , an y f ( x , y ) w ith p r o p e rtie s
(1
5 •1 )
“ ( 1 5 . 6 ) must be th e f u n c t i o n g iv e n by
th e fo rm u la ( 1 5 * ) P r o o f:
By ( 1 5 . 1 ), f o r any y )>
F o u r ie r t r a n s fo r m o f f ( x , y ) ;
we ca n form th e
4>(oc,y).
l e t i t be
t h a t i f f ^ - /^ i *1 1^ -norm , th e n (s e e 1 . 5 ) ) .
0,
4>n (oc)
We know
cj>(c
6 (oc),
+0 , f o r e v e r y ex .
as y
A g a in , from ( 1 5 *3 ) and th eorem 3, ( 15. 7 )
T [ ^ - | ] = ( - ioc )2 d)((x,y); ox
fu r th e r m o r e , mr f ( x , y + h ) - f ( x , y ) 1 _ d)((oc).
By t h e u n iq u e n e s s theorem f o r t r a n s f o r m s i t f o l l o w s t h a t f(x,y)
i s the sta te d fu n c tio n ( 1 5 * * ) .
Rem arks:
The f u n c t i o n s
( 15-9)
( 15 * ) and ( 1 5 * * ) a r e such t h a t
lim f ( x , y ) = f ( x) y -> 0
a lm o st e v e r y w h e r e , f o r ea ch o f them.
However theorem 21
i m p l i e s t h a t a t e v e r y p o i n t x a t w h ich ( 1 5 - 9 ) e x i s t s f o r ( 1 5 * * ) i t a l s o e x i s t s f o r ( 1 5 * ) , but p erha ps not c o n v e r s e ly .
In o t h e r w o r d s , f o r a g i v e n boundary f u n c t i o n f ( x )
in L1 the s o lu tio n (15*) o f the h e a t-e q u a tio n b 2u
6u
n °
’
i s more s t a b l e t h a n t h e c o r r e s p o n d in g s o l u t i o n ( 1 5 * * ) o f the w ave -eq u a tio n
§16 -
Mean v a l u e s
6 2u 6x 2
^7
b 2u
6t 2
+
°'
(w ith t w r i t t e n in s te a d o f the p re v io u s y ) ,
§16.
Mean values
Let q y 0 , and l e t 00
SR=
$
a(oc)K(|)doc ,
R^ ,
Rq
CO
$
K («)dA(oc)
o
K
where A(oc) =
(X
C a ( x )dx o
THEOREM 25 . (1 ) K(oc) is. defined and ab s o lu te ly con tinuous in 0 {(x < go , (2 ) K(y)y^-^ 0 as y - ) oo, and oo
00
$ |K(oo t q
C o n clu sio n : l im R-^oo
y
gd
Rq
.
|K ’ (oc) lex
{ a(oc)dcx = c
.
o
e x i s t s f o r e v e r y R and S-p = c . q K
00
n -1
C K(oc)( b j+ t , x < a . - t
(x) = b .-x+ t -----, b j < x < b j+ t x + a .+ 1 ■ i , a . - t
< x < a . .
Let
As we h a ve o b se r v e d i n C h a p te r I , u / ( o c ,) = 0 ( —-5—), T J oc d
§6, as
ex. J
oo
and i s bounded i n E 1 , and so
y (tx. •) £
( 2 .1 )
L1
i n -00
< a ■ < qd ;
h ence
(2 .2 )
b a j ' bj
J
1 = 2 rr 2ff
r°° I , -00
doc. .
Now d e f i n e (
2.3)
? l U i , . . . , x k ) = J g a .,b .U j) ,
where I i s t h e p r e v i o u s l y i n t r o d u c e d b o x , and th u s g j v a n i s h e s o u t s i d e a l a r g e r box whose p o s i t i o n i s d e f i n e d by I and I
.
j(oc 1 , . . . theorem , (2 A )
L e t t h e F o u r i e r t r a n s f o r m o f g j ( x 1 , . . . , x ^ ) be )•
By t h e o b s e r v a t i o n j u s t p r e c e d i n g t h i s
§3. S u b s tit u tin g
61
Gauss summMlity formula
( 2 . 2 ) i n ( 2 . 3 ) and th e n u s in g ( 2 . k ) , we o b
ta in (2.5)
g*(x) = -L
where
L
V j(o c )e
- i y~ix .x . J J dV
rC
y j(c x ) b elo n g s to L1 in
on a c c o u n t o f ( 2 .1 ).
Now,
owing t o t h e a b s o l u t e c o n v e r g e n c e o f t h e i n t e g r a l i n ( 2 . 5 ) we deduce t h a t S-p f ( y ) g t ( x - y )dVy = L , d>(ocXi;t (oc )e \ I \ YI S i n c e by a s s u m p t io n ,
4>(cx)
J' J'dV («1 , . . • > % ) • We have s e e n t h a t
62
II.
FOURIER TRANSFORMS IN L1 (SEVERAL VARIABLES)
where
.
03 - t x r; 'K ^ r ) =
$
“ 00
• e
dxr
/ -of 2 / ^fc2 J / 2 P r 7 “
- 1^ by ( 7 - 1 0 )C h a p te r I .
e
7X
Hence
k/2 -(Of 2 + . . . + 0C 2/ 4 t 2 ) ( K o c , , . . . , ^ ) = ■=-{E- e 1 k S e t t i n g £- = R, we h ave ( 3 .1 )
T [e
-H x
J
2/R2
kk/2
] = R ir '
e
-R2 ^
- 2 )/* J
D efin e
1^
G . r - i^ jx jX • ~ Y ( 3 . 2 ) s ( x ) = ------ ; i p (Hoc, , • • • jOfi^Je J J .e R (2 n f
dV *
By t h e c o m p o s i t i o n theorem (T h .32)a n d ( 3 . 1 ), we h a v e : G k/2 k S (x ) = ^ , R R ( 2tt)K
(3 .3 )
f(x+ y)e
-R2 £
j
12 ) / 1' J
^k
A lso , ( 3 • *0
A . . S (x)- f(x)= R
Rk
r
...
-k ^ y g i-g,
2 tV
w&
.
. . ..
ff(x + y )- f ( x ) i e
-R2 ( Z > i 2 ) A
sin ce
-.2, 2
k/2^ n ' - R_ \
(2„>k
dV„
Ek:
X
e
- R2 ( Z y i
J
)/h
r°°4-k:-i
dV = 11-----r~“ \ y (» )k
where 2TTk / 2 = -------k 1 r(k/2 )
Wi__,
7rk /2Rk
t
"R 1
w, ne k" ’
/k ■
c
§3. i s t h e k-1
63
Gauss summability formula
d im e n s i o n a l volume elem en t o f t h e u n i t - s p h e r e :
y 1 2+ • • • + y k 2= 1 and so k / 2 Rk . -R2 ( S I y , 2 )/^ Rk 00 k _ -R2t 2/U -----F" Jp e dVv = "V-V ------ 5 t e dt (2rr) k y 2 r(k/2 ) o
2 r(k/2) S
k-1 x
~X^ e
dx
= 1 . N ext, d e fin e
'k -1 2
2
where cr I s t h e u n i t - s p h e r e y 1 + . . . + y^ = 1 , do” (k-1 ) d im e n s i o n a l volume e l e m e n t , and w^._1 i s d i m e n s i o n a l volum e. (3 .6 )
gx ( t ) = t k_1 t f x ( t )
Then, s i n c e
P
F u rth e r m o r e ,
R Wi .
(k-1 )
set - f(x )].
( 3 . * 0 ca n be w r i t t e n as
CD Vr_ 1 _ p 2 i 2
S p ( x )- f ( x ) = - i r f y i J 2 rt /
we have
its
is it s
o
/1
t K ] e * L /4d t
f [f(x + ty)-f(x)]d cr cr
2
64
II.
FOURIER TRANSFORMS IN L1 §4 .
THEOREM
35.
(SEVERAL VARIABLES)
Gauss summability theorem I f f ( x 1 , . . . ,x^ ) C L] in E^ and gx ( t ) is
defined as in (3*6), and i f H(t), 0 t < oo is_ such that j£ t H ( t ) decreases monotonely to ze ro , thus also implying H ( t ) ^ 0 and H ( t ) =
0(
)
as_ t -)> oo , then fo r a fixed x,
the condition t
,
Gy(t) = 5 gx(y)dy = o ( t ) as t -> o x
o
x
implies k r00 R S g ( t ) H ( R t ) d t = 0(1 ), o
as R ^
co
.
THEOREM 56. I f f ( x 1 , . . . ,x^ ) s a t i s f i e s the assumptions of theorem
55
and H is_ such that there e x is t s an Hq with
the property |H (t) | ^ HQ( t ) and Hq now s a t i s f i e s t he a s sumptions on H in theorem
55,
then the condition
t , J IgyltJ Id t = o (t ) o
as t
-> o
implies , H
00 [
o
gv (t)H(Rt)dt = 0(1) x
The proofs of theorems
55
as_ R
00
.
and 36 are very sim ilar to
those of theorems 6 and 7. A ctu a lly the conditions imposed on H(t) are at present somewhat
more s p ecial than those imposed
t h is accounts
previously, and
fo r ace rta in amount of s im p lific a tio n
in
the wording of the present theorems, i f not t h e ir proofs. Using these theorems and the formula ( 3 *7 ) we deduce
§4.
65
Gauss summability theorem
that G S (x) R
f(x )
a s R - ) 00
w hen ever
5t
gx ( t ) d t = o ( t
lr
)
as t
-)
o
Now i f f ( x ) £ L 1 t h i s everyw here;
it
C1* - 1 )
l a t t e r co n d itio n is
i s a s p h e re o f r a d i u s £
V(S^ ) i s t h e volume o f most a l l x Q i n th eorem .
s a t i s f i e d a lm o st
i s e q u iv a le n t w ith s t a t i n g th a t
f ( x o ) =£ ^ m0v ( ^ T
where
0.
.
h t r{x)dYx
>
w i t h c e n t e r a t x Q, and
The v a l i d i t y o f
( U. 1 ) f o r a l
is th en p a rt o f the L e b e s g u e - V ita li
Hence we have
THEOREM 3 7 . The F o u r i e r t r a n s f o r m o f a L ebesgu e r a t e g ra b l e f u n c t i o n i s G a u ss- summable a lm o st e v e r y w h e r e . More e x p l i c i t l y : 1 r lim ----- j- L , ( ------- 5- j y (2n)
L ^k
I (oc) a l s o depends o n l y on t h e d i s t a n c e vex1 + ...+(0C) =
(2T7-)2
$
1 f ( t ) t 2 ( | ) 2 3 1 ^ tC>C dt
o 00 oc cf)((x) = brr ^
f (t )t sin toe dt .
S e t t i n g ([)((x) =oc(t(oc) and P ( t ) = t f ( t ) we g e t : (7 .12 )
_ $>(*) =
00 £ F (t)
sin oct d t .
76
II.
FOURIER TRANSFORMS IN L 1 ( SEVERAL VARIABLES)
I n v e r t i n g ( 7 - 1 1 ) we g e t ( 7 -1 3 )
f(t) =
00 ^ (
a s n ->
oo .
a t g , t h i s would im ply t h a t
-/’ 0 a s n oo , w hich i s
im p ossib le
II T f || II Tg - Tg|| = ------ 2 —
>
1
sin ce ;
M J I f n ll
t h u s we h a ve t h e f o l l o w i n g THEOREM b-k. F or l i n e a r t r a n s f o r m a t i o n s t h e c o n c e p ts o f boundedness and c o n t i n u i t y a r e e q u i v a l e n t . The bound o f a l i n e a r o p e r a t o r T i s d e f i n e d t o be ITI f£
l.u .b Bf
I' T f " I^H
We n e x t p r o v e a n im p o r ta n t theorem on t h e co n v e r g e n c e o f a se q u en ce o f l i n e a r o p e r a t o r s . THEOREM k 5 . I f
|Tn l is_ a se qu en ce o f c o n t in u o u s l i n e a r
o p e r a t o r s d e f i n i n g t h e t r a n s f o r m a t i o n s (f> = Tnf o f t h e sp a ce
i n t o B^, and t f
|Tn l is_ u n i f o r m l y bounded ( t h a t
i s , ITn I = Mn < M f o r a l l n ) , and i f f u r t h e r , Tng c o n v e r g e s t o an o p e r a t i o n Tg on a l i n e a r s u b s e t where d en se i n B^, t h e n f o r e v e r y f
|g| w hich i s e v e r y
i n B^ t h e seq u en ce Tnf
c o n v e r g e s t o an o p e r a t i o n T f and T is_ a l i n e a r c o n t in u o u s o p e r a t o r such t h a t P roof.
|T| ^ M.
I f f and g a r e a r b i t r a r y e le m e n t s o f B^, we
h a ve I! V
V
l! < 11 V
' V
11 + 11
+ 11 TmS-Tmf 11 •
90
III.
S in c e
Lp-SPACES
[Tn i c o n v e r g e s on fgi we h a v e , by C a u c h y - c r i t e r i o n ,
lim || T g -T g|| = 0. m,n->co n m
Futhermore
I! Tnf - T ng|| = II Tn ( f - g ) | | < M j l f- g | | < M || f - g
||
and s i m i l a r l y
11 T me ' Tmf
11 ^ M 11 f " g
11 •
Hence Tim || T f - T f II < 2M || f - g m,n^QD n m Sin ce
fgi
|| .
i s ev ery w h e re dense i n B.^, || f - g
s m a ll a s we p l e a s e .
II ca n be made as
T herefore
H i || T f m,n->oo n
~ T f m
11=
0 .
S in c e B^ i s c o m p le te , t h e sequence Tnf has a l i m i t T f i n B^.
Now t h e o p e r a t i o n T f i s l i n e a r , T (a f) =
lim T ( a f ) = a . lim T n-^oo n-^oo
and s i m i l a r l y T ( f ^ bound M, f o r ,
f 2 ) = Tf 1+ T f2 .
b e c a u se f = a . Tf
F i n a l l y , T f has t h e
g i v e n £ > 0, i f n i s l a r g e enough || T f ||
i s
If
jg i ch o se n .
th e Now
The T so extend ed i s o b v i o u s l y l i n e a r ;
i s bounded, b e c a u s e II T f || - || Tgn " T f || < || Tgn ll < || T f || + || Tgn - T f || or, I II Tgn » - II T f || | < || Tgn - T f ||
0
it
§4.
Linear operations
93
and I M II gn ll - M || f n l| I < M || gn - f || < 6n -> 0 so t h a t II T f || - £n < || Tgn ll < M || gn !| < M || f || + 6n and hen ce II T f || < M II f II . The u n iq u e n e s s o f t h e e x t e n s i o n f o l l o w s from t h e f a c t t h a t the lim it
i n a B - s p a c e i s u n iq u e , and from t h e f a c t t h a t
f o r T t o be c o n t in u o u s i t
i s n e c e s s a r y t h a t Tgn ~^ T f
w h en ever gn ~^ f • THEOREM
48.
I f the tra n s fo rm a tio n T d efin ed in
theorem 47 is_ such t h a t (f o r t h e ex ten d ed T)
|| Tg|| = M || g|| f o r g £ f g j ,
|| T f
11 =
then
M || f || f o r a l l e le m e n ts
f £ B1 . P ro o f: I
II Tgn l| - || T f ||
i
ii
| < || Tgn - Tf||
0
and gn ii - i i f
ii
i
b). We s h a l l now p ro v e two fu n d a m en ta l i n e q u a l i t i e s f o r fu n c tio n s in L ^ (a,b ). THEOREM
49.
(H o l d e r 1s I n e q u a l i t y )
I f f (x ) € Lp (a , b ), p > 1 , and g ( x ) £ Lq ( a , b ) t h e n f ( x ) . g ( x ) e L 1 ( a , b ) and b
b b f (x ) g ( x )dx | < ( £ If (x ) |pdx )p ( § | g ( x ) | qd x ) q . a a a P ro o f:
We may assume t h a t f + 0 and g + 0.
C o n s id e r
the fu n c tio n
f o r t ^ o.
We have f ( l ) = f 1 ( 1 ) = 0; f u r t h e r m o r e , f ’ ( t ) > 0 ,
f o r 0 < t < 1 , and f ’ ( t ) < 0 f o r t > 1.
Hence f ( t )
f o r a l l t y o e x c e p t f o r t = 1, when f ( t ) = 0.
( t h e e q u a l i t y h o l d i n g o n ly f o r t = 1 ) . t = | A | . |B|
1
-n
Q
S e ttin g
and m u l t i p l y i n g by B , we o b t a i n
Thus
< 0
§5*
Lp-spaces
95
(5-1 ) I f we ch o ose
a
a
t h e n we h a ve a
a
now t h e p r o d u c t A .B i s m e a su ra b le (on a c c o u n t o f a w e l l known p r o p e r t y o f m e a su ra b le f u n c t i o n s ) , and s i n c e t h e rig h t sid e o f ( 5 -1 ) is in te g r a b le , is
i t fo llo w s th at
|A.B|
Hence i n t e g r a t i n g ( 5 * 1 ) we g e t t h e r e
in teg ra b le.
quired r e s u l t b
THEOREM
50.
b
(Minkowski 1s I n e q u a l i t y )
I f b o t h f ( x ) and g ( x ) b e l o n g t o L p ( a , b ) , p ^
1
, th en
we h a v e :
a
a P ro o f:
We may suppose t h a t f ( x ) ^ .o and g ( x ) ^ o .
Now
a
a
a
A p p l y i n g t h e i n e q u a l i t y o f t h e p r e v i o u s th eo rem , we o b t a i n
III.
Lp-SPACES
j V + g l P dx < ( $b | f + g ! P d x ) q . (^b | f | p d x ) p a a a
a
a
w h ich i s t h e r e s u l t we r e q u i r e . I f we now i n t r o d u c e t h e q u a n t i t y 1 < p
t h e n , by theorem 50 i t has p r o p e r t i e s norm ( s e e § 3 ) , bu t not p r o p e r t y (1 ),
o ,
j , te > j
0 (t),
§5 * Lp-spaces
97
t h e n t h e r e e x i s t s a su bsequ en ce f e v e ry w h e re t o some f ( x )
(x ) c o n v e r g i n g a lm o st nn i n L p ( a , b ) , and 1
($ I f m“ f l P d x ) p -> 0
a s m -> 00 .
A l l t h e above p r o p e r t i e s o f L ^ - s p a c e s h o ld i n g e n e r a l t y p e s o f measure s p a c e s ;
in p a r t ic u la r ,
ces o f k-dim en sion s, k ^ 1
, and we ca n o p e r a t e w i t h any
m e a su ra b le s e t A i n s t e a d o f bounded i n t e r v a l
in E u clid e a n spa
the i n t e r v a l ( a , b ) .
( a , b ) ---- and more g e n e r a l l y ,
For a
f o r a set
A o f bounded m e a s u r e ---- a f u n c t i o n g ( x ) o f c l a s s L ^ ( a , b ) , p )> 1 , i s a l s o a n elem ent o f L 1 ( a , b ) . H o ld e r’ s in e q u a lit y to the fu n c tio n s b f Ig ( x ) |dx < a
In f a c t |g(x)|,
i f we a p p l y
1 , we o b t a i n
b
b I g ( x ) |p d x ) p .($ 1 q dx )^= ( b - a ) q || g|L • a a p
More g e n e r a l l y , f o r 1 ^ p< p ,
any f u n c t i o n o f
L p ( a , b ) i s a l s o a n elem ent o f c l a s s L p , ( a , b ) . no l o n g e r t r u e , f o r i n s t a n c e ,
in the i n t e r v a l
cla ss But t h i s
is
(0,oo ), th e
L ebesgu e measure o f t h e i n t e r v a l b e i n g i n f i n i t e . Now, t a k e a f i n i t e o r i n f i n i t e more g e n e r a l l y a s e t A ) .
in te r v a l (a ,b )
A fu n ctio n f ( x )
in i t
(or,
is c a lle d
f i n i t e l y v a lu e d i f t h e r e e x i s t s a f i n i t e number o f d i s j o i n t m e a su ra b le s u b s e t s A 1 ,
...,
A
o f ( a , b ) ea ch o f f i
n i t e m easure such t h a t f ( x ) has a c o n s t a n t v a l u e c^ on A^, k = l , . . . , n and t h e v a l u e z e r o on t h e s e t w h ich i s co m ple m entary t o A 1 + . . . + An , i n ( a , b ) .
Now i t f o l l o w s from g e
n e r a l p r i n c i p l e s o f L eb esgu e measure t h a t f o r ea ch p , 1 ^ p 0 -oo
as h
0; more g e n e r a l l y , P ro o f.
P la in ly ,
(h ) is_ c o n t in u o u s i n h .
r f (h) < 2|| f|| .
G ive n
t
> o th ere
§6 . Continuity,summability and approximation in I^norm 99 e x is t s a continuous function 6
1
k
I f we now assume t h a t H(cx) i s a r a d i a l f u n c t i o n , t h e n we o b tain :
11 . I
2
oo < c Rk ( f
6
= c Rk (
5 °°
1 — |H(xR)|q x k_1 d x ) q
| H ( y ) Iq y k - 1 R ' k d y ) q
6r
k ( 1 " n ) r00 a k-i = c R q ( [ IH (y ) |q y 6r oo
—
= c Rp ( f
a dy)q 1
| H ( y ) |q y k_1 d y ) q . 6r
I f we assume f u r t h e r t h a t H(
111
) =
0 ( ---- 1— f ) ,
fc. > o
It|k+t '
then i t fo llo w s th at
11 2 I
= o( 1 ),
A gain , I I , I = o( i )
a s R -> oo
r
§6. Continuity,summability and approximation in Lpnorm 103 on u s i n g t h e same argument a s i n theorem s 6 , 7 , 3 5 and 36, p rovided th a t x k H(x) - ^ 0 ,
a s x - ) 00
w h ile t \ g „ ( t )dt = o ( t ), a s t -> o , o x where gx ( t ) i s d e f i n e d a s i n §3 o f C h a p t e r I I .
We a r e
thus i n a p o s i t i o n to s t a t e the fo llo w in g THEOREM (i) (ii) (H i) (iv )
55.
If
f(x) €
P > 1
H(cx-) i s a r a d i a l f i m c t i o n . and H(oc) ^ 0, — S? (2-n-r
S tt k
= 1
H( lex I ) = 0 ( — lex I
) as
00 ’
t > 0 ,
th en the c o n d itio n t 5 gx ( t ) d t = o ( t ) , o
as t
-> o
im p lie s S p ( x ) - f ( x ) = 0(1 ), Remark: ( i i ) and ( i i i )
as R ^
oo .
t o g e t h e r im p ly t h a t H(oc) € L 1 i n E^.
1 ok
CHAPTER IV FOURIER TRANSFORMS IN L,J2§1.
T r a n sfo r m a t io n s i n H i l b e r t sp ace
I n t h i s c h a p t e r we s h a l l be concern ed w i t h F o u r i e r t r a n s f o r m s i n "L . f(x)£
S in c e , f o r an a r b i t r a r y f u n c t io n
L2 (-od,oo) t h e i n t e g r a l d e f i n i n g t h e F o u r i e r t r a n s
fo rm
0,
t h e r e e x i s t s a 6 such t h a t 5( l l f 0 ll + II g0 ll) + 62 < t
and t h e n I(f,g)
- ( f 0 , g Q )I < t
fo r II f - f 0 II < 6 ,
and
|| g - g QII < 6 .
I f S 1 and S 2 a r e two s u b s e t s o f L2 (-qd ,oo ), a t r a n s f o r m a t i o n T o f S 1 i n t o S2 i s c a l l e d
iso m etric i f f o r every
p a i r o f e le m e n ts i n i t s domain, we h ave ( T f , Tg) = ( f , g ) . T is c a lle d u n itary i f L2 , and i f
i t s domain and ran ge c o i n c i d e w i t h
( T f , T g ) = ( f , g ) f o r f , g € L2 .
§2 . Our o b j e c t
P l a n c h e r e l f s th eorem
in th is
s e c tio n is to d efin e a F ou rier
t r a n s f o r m f o r f u n c t i o n s i n L2 , and t o show t h a t i t
repre
106
IV.
FOURIER TRANSFORMS IN L,2
se n ts a u n ita r y tra n sfo rm a tio n in h .
In o r d e r t o e x h i b i t
c l e a r l y the i n v e r t i v e p r o p e r t ie s o f the F o u r ie r tran sfo rm i n L»2 , i t
i s c o n v e n ie n t t o a l t e r t h e c o n s t a n t s o c c u r r i n g
in our d e f i n i t i o n s
in the L 1 -th e o r y .
H e r e a f t e r we s h a l l
s a y t h a t i f f ( x ) £ L ] (-00,00 ) t h e n i t s F o u r i e r t r a n s f o r m cfc>( o c ) i s d e f i n e d by:
(2. 1 ) W ith t h i s d e f i n i t i o n i n mind, we s h a l l c o l l e c t h e r e two o f t h e r e s u l t s we need from t h e L 1 - t h e o r y .
I f S i s the c la s s
o f bounded f u n c t i o n s b e l o n g i n g t o L 1 (-00,00 ), t h e n S i s a d ense s u b s e t o f L2 ( -o o ,o o ), and i f f ( x ) € S, t h e F o u r i e r t r a n s f o r m i s d e f i n e d by t h e i n t e g r a l on th e r i g h t s i d e o f ( 2. 1 ), w h ich e x i s t s f o r e v e r y cx i n - < cx < 00 .
We t h e n
have, f i r s t : i f f , ( y) | -00
GO
£ I f ( x) | -00
(See theorem 1 2 ) .
F in a lly ,
f o l l o w s , f o r b > a and
t
i f ^ ^( y )
> 0:
dy
d e f i n e d as
§2.
Plancherel*s theorem 0,
y < a-t
1)
a < y < b
y - a + 5.
107
a-t < y < a
t
b+e -y t o ,
b < y < b+ t y > b+e.
t h e n t h e l i n e a r m a n ifo ld D d ete rm in ed by f u n c t i o n s o f t h e form d)^ k
d ense I n L2 , b e c a u s e i t s
clo s u re c o n ta in s,
in
p a r t i c u l a r , the s t e p - f u n c t i o n s . Now, d e f i n e f^ K ( x ) cx
(2.U)
the e qu atio n :
jD
4 , b (x) = i/Sf C ^ ' b(y)6"lyX d y '
Then, o b v i o u s l y , f
9#^>.(x) D
i s a bounded f u n c t i o n o f c l a s s L - , 1
and i n p a r t i c u l a r (by ( 2 . 3 ) ) we h a v e : (2 .5)
dx
^a,b
a lm o s t e v e r y w h e r e , and OO
5
-00
c
OO
,< M( yy )tyyaaj ,tb| (( yy ))dd yy
ill
oo = = i$Q0:f (x )ga > b ( -x )dx
or (2 .16 )
b oo $ 4>( y) dy = = $ a v 2tt -00
ibx_
7
f o r any f £ h .
iax
— ~ “ —
dx
ix
T h is may be c o n s i d e r e d a s t h e second
in te r p r e ta tio n o f r e la tio n (2 .7 ). I f on t h e o t h e r hand, f € L 1 n L2 , t h e n from ( 2 . 1 5 ) we h ave t> - 0 0 b . $ (y) = - 1— n
1
t h e n ea ch f n ( x ) £ L 1 n >?_, and f n (x ) -> f ( x ) Hence, by ( 2 . 6 ) and ( 2 . 1 7 ) we have n
(2 .18 )
J) = T f = ~ n n
5Jnff ( x ) e l y x -n ~
dx-
i n L2 -norm.
112
IV. FOURIER TRANSFORMS IN L2
T h is i s t h e t h i r d
in te rp re ta tio n of r e la tio n (2 .7 ).
I f i n ( 2 . 1 6 ) we choose a = o and b = t , we h a ve: t 00 0itx_ 5 ^(y)dy = — 5— w — f (x)dx. Jo V/2rr -00 1X Hence f or almost a l l t , (
2 . 19)
1 (y ) € L2 , and the t rang formation T defined by Tf ( x ) = (My) = l . i . m . y — ^ f(x)e^~yx dx n-^00
is a unitary transformation of L
V 2 tt -n into i t s e l f .
Similarly,
§3.
General summability
113
the functions v n
( y ) = — :r $ g ( x ) e ~ l y x d x , \/2u -n
co n v e r g e i n L2 - norm t o a f u n c t i o n tra n sfo rm a tio n T
g(x)£
L ,
2
y (y ) € L2 and t h e
d e f i n e d by
T *g = y ( y ) = l . i . m $ g ( x ) e ' i y x dx n-^oo V2rr -n i s t h e i n v e r s e o f t h e t r a n s f o m a t i o n T, so_ t h a t wh en ever the r e l a t i o n (y) = l . i . m . —— £ f ( x ) e ^ x dx n-^oo \Z2rr -n h o l d s , t h e n so d oes t h e o t h e r : f ( x ) = l . i . m . -zzz $ oo ,
§3. ( 3 .7 ) is
General summability
4>y ( t ) = !k( y + t ) ^ ( y - t )
115
_
such t h a t
i
t
r- ^ ( t ) d t o ^ R em arks: I f
o
as t
- ) o.
|H(y)| ^ HQ( y ) , where HQ( y ) now s a t i s f i e s t h e
co n d itio n th at H s a t is f ie d t
i n theorem 58, t h e n
5 Ig ( a ) | d s = 0 ( 1 ) o
as t - ) 0
w i l l im p ly (3 A ) , w h i l e 1 r- ^ | ( s ) | d s = 0 ( 1 ) o ^
as t - ) 0
w i l l im ply ( 3 . 6 ) . If I 1 - It I , K(t) = \ I 0 ,
It|
< 1
|t|
^ 1
then H(y) =
and Hn ( y ) -
1
and h en ce we h ave ( 3 -8 )
lim k R->oo K
a lm o st e v e r y w h e r e , where
R C (y)d p = (y),
For, i f f n (x) = f( x ) K n ( x ),
y£
B . J
then f n (x)-> f ( x )
And Sn ( y ) = T f n ( x ) , where T i s c o n t i n u o u s .
sn (y)
i n L2 -norm.
That i s ,
Snk ^y )
i n L2 -norm
Hence (i)(y) a lm o st
§4. everyw here.
Several variables
In p a r t i c u l a r ,
S (y) co n verges.
117
on t h e m e a su ra b le s e t B ,
Hence
lim S ( y ) = (y) , n-^oo
y £ B
. ^
Remark: Note t h a t
( 3 . 1 0 ) does no t a s s e r t t h a t lim S ( y ) n-^00 e x i s t s , but on ly says t h a t , i f the l im i t e x i s t s th en i t i s equal to the tran sfo rm .
§U.
Several v a r ia b le s .
The e x t e n s i o n o f P l a n c h e r e l ’ s theorem from one t o s e v e r a l v a r i a b l e s does not i n v o l v e any new d i f f i c u l t y
in
p rin cip le . We s t a r t w i t h t h e f u n c t i o n (|)^ ^ d e f i n e d b y :
where ea ch f a c t o r cj)^ bed i n §2.
p
J
^ (y .) i s e x a c t l y o f t h e form d e s c r i J
Then a t)( y 1 , . . . , y ^ ) i s a bounded f u n c t i o n b e
l o n g i n g b o th t o L 1 and L2 , and i f we t h e r e f o r e d e f i n e c
.
00
00
,
-iy~y-X ■
t h e n , o b v i o u s l y , f*“ , i s a g a i n a bounded f u n c t i o n b e l o n g i n g cl y U t o L 1 and L , and by theorem s 38 and 39 we h a v e , ,
,b(y i
*
(4.3 ) everyw here.
00
00 t
, y k> - ^ / 2L (277-)“-' -oo - - -Lc d f a ,b < x i ’ - - - ' x k )e
Fu rth erm ore
iZ x ^ y .;
1 1 8 IV.
FOURIER TRANSFORMS IN L„. co
oo
-O O
“ OO
loo ( 2tt) '
In p a r t ic u la r ,
,
2
=xi
2
2
+««*+x^ , t h e n we o b t a i n
(... f
f(x)e
X l 2+ . . . +V
(y^ , . . . , y ^ ) £ L 2 and
the tra n s fo rm a tio n T d efin ed b y : T f (x ) = co
------ j ( 2 tt)
[ ----- f
^
i ^ j c -y • 3 JdV x
f(x )e
2< ^2
h o l d s , t h e n so does t h e o t h e r : f (x ) = l . i . m . n^oo
i -i-X ^ x . ------ rr/o ?•••$ (y)e J JdV ( 2 tt) ' " H x .2< n2
and ^ | f ( x ) | 2dvx =
§5 •
^ l < t > ( y ) l 2dvy
R ad ia l fu n c tio n s
Assume t h a t f ( x ^ , . . . £ d efin e d in C h r .I I ;
.
L2 i s r a d i a l i n t h e se n se
then
f ( x 1 , . . . , x k ) = f C Ix | ) = f (v/x1 2+ . . .
+ x k 2 ).
Set I f n (lx | ) = 1 i
f ( |x)), 0
,
fo r
for
Ix| ^ n.
Then f n ( x 1 , . . . , x k ) -> f ( x 1 , . . . , x k )
|x| < n
122
IV.
FOURIER TRANSFORMS IN Lg
i n L2 -norm, where f n £ L 1
0
L2 .
Sin ce f
£ L1 , i t fo llo w s
t h a t 4>n (y) i s a l s o r a d i a l , and by P l a n c h a r e l ' s theorem ,
4>(y)
6n ( y )
= T f.
S in c e t h e r a d i a l f u n c t i o n s i n L2 form
a c l o s e d su b sp a ce o f L2 , i t f o l l o w s t h a t ct>(y ) i s a l s o r a d i a l (and s i m i l a r l y f o r t h e i n v e r s e ) . (5.1 )
Thus,
i f f £ L2 i s r a d i a l , t h e n t h e F o u r i e r t r a n s f o r m o f
f is a ls o r a d ia l. Now, i f f f(t)tk 1 t
is r a d ia l,
t h e n f £ L2 (E^) i f and o n l y i f
L2 ( 0 , oo ), where f ( t ) = f (\Zx~ 2+ . . . +x^.2 ). F ( t ) = f ( t ) t k_1
, _
© t ) = ^ ( t ) t k_1
.
Set
and
As b e f o r e , we t h e n have f o r F ( x ) £ L2 ( o , qo ), _ $.(y) = l . i . m
00 $ F (x)S (yx)dx , o P
where (D(y) £ L2 (0,oo ), and F( x ) = l . i . m . $
00 _
£ (y)S (yx)dy
o
r
and cd _ oo_ _ ^ F G dx = J | y dy o o where k-2
p = —
•
Thus t h e i n v e r s i o n fo r m u la s i n k - v a r i a b l e s ,
i f a p p lie d
t o r a d i a l f u n c t i o n s , produce t h e s o - c a l l e d Hankel i n v e r s i o n
§5. fo r m u la s f o r p. = i n L2 -norm. fo r
p. =
k ~2
123
Radial functions
, and we a l s o o b t a i n t h e i r v a l i d i t y
T h i s manner o f d e r i v a t i o n , g i v e s t h e n o n l y an i n t e g e r or
p. =
h a lf-in te g e r.
However,
t h e fo r m u la s and t h e i r v a l i d i t y rem ain i n f o r c e f o r a r b i t r a r y r e a l valu es o f
p.
, as we w i l l show i n t h e n e x t
chapter. I f k=l t h e Hankel t r a n s f o r m r e d u c e s t o t h e c o s i n e t r a n s f o r m , w h i l e k=3 g i v e s so m ethin g v e r y a k i n t o t h e s in e - t ran sform . I f k = i, a r a d ia l fu n ctio n is
j u s t an e v e n f u n c t i o n ,
and h en ce a s i n ( 2 . 1 9 )
J
a lm o st e v e r y w h e r e .
The r i g h t s i d e I s t h e d e r i v a t i v e o f an
I n t e g r a l o f an L 1 fu n c tio n , where.
o
so (y ) e x i s t s a lm o s t e v e r y
F o r t h e i n v e r s e , we h a v e : OO o
J
a lm o st e v e r y w h e r e .
§6.
D e r i v a t i v e s and t h e i r t r a n s f o r m s
We w i l l now have some o p e r a t i o n a l F o u r i e r t r a n s f o r m s i n L2 .
theorem s i n v o l v i n g
We w i l l d e a l e x c l u s i v e l y w i t h
t h e o n e - d i m e n s i o n a l s i t u a t i o n , a lt h o u g h t h e k - d i m e n s i o n a l a n a lo g u e s a r e f r e q u e n t l y o f c o n s i d e r a b l e i n t e r e s t and f a r from t r i v i a l .
124
IV.
FOURIER TRANSFORMS IN L2
Some o f t h e theorems t o f o l l o w a r e e x t e n s i o n s m ain in g o n e - d im e n s io n a l
(re
) t o L2 - f u n c t i o n s o f theorem s p r e
v i o u s l y s t a t e d and proved f o r L 1 - f u n c t i o n s .
In some such
i n s t a n c e s , th e L 1 - c a s e i s t h e more d i f f i c u l t
one, and i t
o f t e n r e q u i r e s s e v e r a l e l a b o r a t e m a n i p u l a t i o n s , where th e L2 c a s e co u ld be d is p o s e d o f i n f e w e r s t e p s ,
the reason
b e i n g t h a t i n t h e L2 ~ c a s e , n o rm -co nvergen ce o f t h e f u n c t i o n can be im m e d ia te ly t r a n s l a t e d
i n t o norm c o n v e r g e n c e
o f th e t r a n s f o r m (and c o n v e r s e l y ) , w hereas i n t h e
-case
no such t r a n s l a t i o n i s d i r e c t l y a v a i l a b l e bu t must be l a b o r i o u s l y s u b s t i t u t e d f o r by o t h e r r u l e s o f p r o c e d u r e . THEOREM 6 0 . I f f ] ( x ) € L 1 i n E ] , and T [ f n ( x ) ] and f
2 (x)
€
in E ^
(Or),
and T [ f 0 ( x ) ] = 2 (oO, and i f
(cx) = 0 (cx) a l m o s t e v e r y w h e r e , t h e n f
1 (x)
= f
2 (x)
a lm o st
everyw h ere.
P ro o f: (6.1)
f
E x ce p t f o r a n u l l - s e t , we h a v e , by theorem 6, ( x)
= lim — ^ ( a ) e -l0fXe " gd , we o b t a i n
r° °
p
j
CX | d>(oc) I
p
dix
< M.
"00
Thus -ioc 4)(oc ) €
( 6.15)
L 2 (-0 0
,0 0
).
S in c e g ^ ( x ) - ^ g ( x ) i n L2 -nonn, we a l s o have -ioch
00
(6.16)
lim
^
h~7> 0
|
— t-
2 (oc)-
uj(oc) |
dc* = 0.
-0 0
B u t , f o r v a r i a b l e h t h e i n t e g r a n d i s m a j o r i z e d by 2 2 2 oc I (Hex) I + ly(or)l w hich b e lo n g s t o L . Hence, we can l e t h-7>0 under th e i n t e g r a l oo ]
s i g n i n ( 6 . 1 6 ) , and t h i s g i v e s
2 I -ioc (t( ^ d z V2rc -oo
f(x,y)
^
K -x2 i s o b t a i n e d from S p (x ) i n ( 3 - 2 ) i f we pu t K(x)==e
and y = . R P ro p e rtie s o f f ( x , y ) : ( 7-1 )
F or e v e r y y )> 0, f ( x , y ) £ [In f a c t ,
its
L
as a f u n c t io n in x . -
t r a n s f o r m (y)e ^
oc 2
b e l o n g s t o L0 , i f
cb(oc) does . ] (7.2)
(7 . 3 )
l i m f ( x , y ) = f ( x ) i n L -norm 0 2 [ I n f a c t , (oOe ^ c o n v e r g e s i n L2 -norm t o
0,
V■ P t h e r e e x i s t s a f u n c t i o n 5^ € L2
such t h a t
^ lim ( 5 ° ° | h->0 -CO
In f a c t , (
|2 d x ) 2 = o . by
h
t h e t r a n s f o r m o f f ^x >y+^-)--£-(i L>y) j_s -vtx2 e “h0( | ^ ^
7A a )
| .
S in c e ~h(X 2 g- — ~1- | < I i ( a ) | e _yDC oc? ,
2
I(oc)e_yc
(oc) ] =
0
in S '.
Thus (Kcx) =
0
0, (Oc)
[ T^tDf (x ) ] h i n L2 ~norm.
By theorem 66, i t f o l l o w s t h a t T ^ D fU )]
i s t h e d e r i v a t i v e o f g ( x ) a lm o st e v e r y w h e r e . THEOREM
74.
I f Tx is[ a bounded l i n e a r o p e r a t o r on
L2 (-qo ,oo ) such t h a t T * f ( x ) = g ( x ) , and i f TX [Df ( x )] = D [Tx f (x )] , w h en ever t h e two d e r i v a t i v e s e x i s t , t h e n T*Rh [ f ( x ) ] = Rh .Tx [ f ( x ) ]
.
P ro o f: L e t ct, y be t h e F o u r i e r t r a n s f o r m s o f f and g oc r e s p e c t iv e ly ; then T ^ . C on sid er a f u n c t io n d efin e d f(x) =
by
f
-A
((tx) = T(oc)
by
y(-oc )
U“ 1 : y(cx) = T F o T )(J)( —Oc ) t o be u n i t a r y i s t h a t ( 3 .2 6 )
|T(0
’
b y lemma 6; and t h e r e f o r e lemma 7 a p p l i e s .
§3 .
T a u b e r ia n th eorem s; a v e r a g e s on ( -oo ,oo )
THEOREM 8 3 . Assum ptions : (i)
y(x) € l p
(i i )
S ( x ) is_ o f bounled v a r i a t i o n i n e v e r y f i n i t e i n t e r v a l , and x+1 I
| d S (y ) | < c , (t) €
Jjp
(ii)
S^a ) is of bounded v a ria tio n in every f i n i t e in terval ( A ^ ------------ 0 < n o f ' l M
(i i i )
i i l
1 N 0, (i v )
f o r 0 ( ^ ( a we can th u s form t h e a b s o l u t e l y convergent i n t e g r a l \
oo (t+ioc) = $ W( t ) t ^ o
. +±0< d t ,
-oo (t), D ( t ) and A s a t i s f y a l l t h e
hypotheses of theorem 91 , then s'(A) ~ A A d>(A) as
A-^oo . P ro o f:
fu lfille d F.j(t),
The a ss u m p tio n s abo ut F ( t ) o f th eorem 91 a r e
f o r two n o n - i n c r e a s i n g c o n t in u o u s f u n c t i o n s
F2 ( t ) o f wh ich t h e f i r s t
is
0 f o r 1 < t < 00 , and t h e second i s 0 f o r i+£ < t < 00 .
1 in 0 ( t { 1 - t
1 i n 0 ( t { 1 , and
The number 1 ca n be c h o s e n so a s t o
make 00 j
IF ( t ) - F ( t ) ! d t o
a r b i t r a r i l y sm a ll.
and
The " i n t e r m e d i a t e " f u n c t i o n
2 0k
VI.
GENERAL TAUBERIAN THEOREMS
f 1, P 0( t ) = f I 0,
0 £ 1
t
£
1
< t < 00
is n o t c o n t in u o u s , b u t i t f o llo w s fro m 00
_____
oo
lim{A(A ) J -1 5 F (J ± )d s '(/i) < lim
|«|>( A ) A |_1 J F 0 ( £ ) d s '( ) i)
lim
Ia 0.
S u b s titu tin g FQ
94.
I f (t)€ $p then
4>(t^)€
Substituting A( t ) f or S(t ^), (t(tk ) f or
4>(t),
W(t) f or D(t^) and A/k fo r A, we get theorem 9k from lemma 11 . C o r o l l a r y t o theorem
9^.
I f k ) 0 and W( t ) i s a c o n t in u o u s f u n c t i o n i n
o o, t k ~1W( t ) = O ( t ~ 1 l o g t~ p ~^),
f o r t = 0, and t = oo ,
and £ W(t ) t k 1 +^cx" d t + 0, o
f o r -oo < oc < oo ,
and
I then
hr
y
00
f
an W(y )l
^
c
’
0
< ^ < 00 ’
§5 . Special cases
205
im p ly
Remark: P u t t i n g W( t ) = e ^ lim y k ^ y->co 1
a
we o b t a i n :
e n^
= A. P ( k )
and
then n
T . a.’r
= A .
T a k in g k = 1 we o b t a i n t h a t A b e l s u m m a b ility o f a sequence o f p o s i t i v e term s i m p l i e s
(C,1)
s u m m a b ilit y .
(see C h r . I ) .
As a s p e c i a l c a s e o f theorem 92, we h ave THEOREM
95.
(i)
I f W( t ) i s a m onotonic non- i n c r e a s in g ;
f u n c t i o n f o r w h ich ! 1 + 0 ( t a ),
at t = 0
1 -a 1 0(t )
at t =
,
qd
f o r some f i x e d a )> 0, and l im t-^ + o
f°° W ( t ) t t o
1 + l o c dt + 0
f o r -00 < oc < 0 and 0 < oc < 00; and i f f o r a s lo w - g r o w in g f u n c t i o n (t ) o f c l a s s ^ , we have 00
XI an W ( | ) | < c ,
o < y < oo ,
2 06
VI.
GENERAL TAUBERIAN THEOREMS
th en the l i m i t - r e l a t i o n 1
0000
n n
/4S, TOO TOO f f aa= = “V“V --A A y1- ^
together t o g e t h e r with w i t h tthe h e one-sided o n e - s i d e d ccondition o n d itio n
((5-9) 5.9)
aan n )^ ' |j (n) 0
im ply
( 5- , 0 > (ii)
^
nW
^
% - A :
i n t h e c l a s s i c a l c a s e , we have 4)(y) 4) ( y) s
11 and t h e n .
t h e a ssu m p tio n s £
£
an W ( f )
S
" < f > •>-> AA, "
an i - E an ^ - I
im ply
Z
an = A ;
( i i i ) h o w e v e r , i f (b(y)-^o as_ y-^00 y-^oo , t h e n ( 5 ..1100)) i m p l i e s first
of a ll 00 E
( 5 . 11)
a
= 0
1
and t h e n by combining; ( 5 . 1 0 ) and ( 5 . 1 1 ) we o b t a i n
[Remark on p a r t
1
00 y~ a ^ m=n+l
lim n-^00
^
(iii).
I n t h e c a s e W( t ) = e ^ T a u b e r ’ s
o r i g i n a l c o n d i t i o n was
-
a n= o ( ^ ) ; t h i s
-A .
i s now b e i n g r e n
dered more s p e c i f i c by c o n s i d e r i n g 0 ( 1 ) t o be a c e r t a i n -B i n s t e a d o f n a =
0(1)
P rc.
London M a t h .S o c .( 2 ) 1 3 ( 1 9 1 4 ) 1 7 4 - 1 9 1 ; E.Landau made t h e assumption^, S = n n S
a .
0(1
) and lim max IS ~S l= 0 where 6->+0 |m-n|+0
lim ( S -S „ ) ^ 0 n