Functional Operators (AM-22), Volume 2: The Geometry of Orthogonal Spaces. (AM-22) 9781400882250

Table of contents :
TABLE OF CONTENTS
Preface
CHAPTER I. FOURIER TRANSFORMS IN L1 (One Variable)
§1. Elementary properties
§2. Riemann Lebesgue Lemma
§3. Convolution of two functions
§4. Derivative of a function and its transform
§5. Inversion formula
§6. Uniqueness of Fourier transform
§7. Summability theorems
§8. Some applications of summability-theorems
§9. Continuity in norm
§10. Summability in norm
§11. Derivatives of a function and their transforms
§12. Degree of approximation
§13. Abel’s theorem
§14. Abel and Gauss summability
§15. Boundary values
§16. Mean values
§17. Tauberian theorems
CHAPTER II. FOURIER TRANSFORMS IN L1 (Several Variables)
§1. Riemann Lebesgue Lemma; Composition:, Convolution
§2. Uniqueness theorem
§3. Gauss summability formula
§4. Gauss summability theorem
§5. Application of summability-theorem
§6. Norms, Continuity, Parseval relations
§7. Radial functions
§8. General summability for radial functions
CHAPTER III. Lp-SPACES
§1. Metric spaces
§2. Completion of a metric space
§3. Banach spaces
§4. Linear operations
§5. Lp spaces
§6. Continuity, summability and approximation in Lp norm
CHAPTER IV. FOURIER TRANSFORM IN L2
§1. Transformations in Hilbert space
§2. Plancherel's theorem
§3. General summability
§4. Several variables
§5. Radial functions
§6. Derivatives and their transforms
§7. Boundary values
§8. Simple type of bounded transformation
§9. Bounded transformations commutative with translations
§10. Closure of translations
CHAPTER V. GENERAL TRANSFORMS IN L2
§1. General unitary transformations in L2 (0,∞)
§2. Watson transforms
§3. Functional equation associated with Watson transform
CHAPTER VI. GENERAL TAUBERIAN THEOREMS
§1. Introduction
§2. Preliminary lemmas
§3. Tauberian theorems; Averages on (-∞,∞)
§4. Averages on (0,∞)
§5. Special cases
NOTES

Citation preview

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