Fourier transforms o f the class ~p

A R K I V F O R M A T E M A T I K B a n d 2 n r 30

C o m m u n i c a t e d 8 April 1953 b y ARNE BEURLING a n d AKE PLEIJEL

Fourier transforms o f the class ~p

B y EDWIN HEWITT

It is well known that the theorem of RIESZ-FIscI~ER and the theorem of

I:)LANCHEREL,

dealing with Fourier transforms of the classes s on the circle and line, respectively, have analogues for other classes ~p (1 < p <2). Thus the theo- rem of YOUNG-HAusDORFF states that if [ is any function on [0, 2 7r] such t h a t

2~ 2~

f I/(x)J'dx<~,

then the numbers cn= 2-~

e~=/(x)dx

have the property t h a t

0 0

(1) Jcnl"< ,

where p ' - p - l ' P and

0

(See for example [5], pp. 189-202.) An analogous theorem, proved by TITCH- MARSH (see [3], pp. 96--107), shows that every function / in ~ , ( - 0%00) admits a Fourier transform of class ~,, with norm in ~,, ( - c ~ , oo) majorized by a constant times the ~ norm of ]. For both of these cases, examples can be given to show that not all sequences of class l~, or functions of class ~p, can be obtained as Fourier transforms of the class ~ , . (See [5], p. 190, and [3], pp. 111-112.) It is the purpose of the present note to show that this phe- nomenon must appear for all infinite locally compact Abelian groups.

Throughout the present note, let G stand for a locally compact Abelian group.

Integration with regard t o a suitably normalized Haar measure on G is indi- cated by expressions such as

(3) f /(x) dx.

G

Fo~ all numbers r_> 1, the symbol ~r denotes the space of all complex-valued Haar measurable functions / such t h a t

(4) fl/(x)

G

571

E. HEWITT, Fourier transforms of the class ~p

under the usual definitions of addition and multiplication by complex numbers.

~ is normed b y

(5) II/I1 = If dx]}

G

L e t ~r162 denote the space of all continuous complex-valued functions on G each of which vanishes outside of some compact set. Let G* be the group of all continuous characters of G, topologized in the usual fashion ([4], pp. 99-100).

The expression (x, y) is used to denote the value of the character y e G* at the point x E G, or, dually, the value of the character x e G at the point y e G*.

For a function / e ~ ( G ) , the Fourier transform T t is defined b y the usual expression

(6) T / (y) = J" (x, y) / (x) d x.

a

Throughout the present note, for every number p > 1, let ~ ' - P A. W E ~ has shown ([4], pp. 116-H7), b y using the convexity theorem of M. RIESZ, t h a t the mapping T of (6) has the property t h a t T/~L~,. (g*) and t h a t II TIlls"-<

< H/lip, for 1 < p < 2. Thus T can be extended by continuity to a linear trans- formation T~ with domain ~ ( G ) and range contained in ~p.(G*) such that:

(7) Tp ( ~ (G)) is dense in ~ . (G*) ;

(8) Tp is linear;

<9) IIr, tll,.<-Iltll,.

(Assertion (7) requires separate proof.) I t follows immediately t h a t Tp is a one-to-one mapping. Our aim is to prove the following fact.

T h e o r e m . If G is a locally compact Abelian infinite group and if 1 < p < 2, then the image T r (~p(G)) is a dense set of "the first category in ~r. (G*), and the functions in ~p. (G*) which are not Fourier transforms comprise a dense set of the second category.

This theorem was suggested by a question raised by I. SEOAL [2]. The proof is based on the following two lemmata.

L e m m a A. L e t G be any infinite locally compact group and let ~ be a number greater t h a n 1. Then there exists a sequence {[,}~1 of functions in

~p (G) such t h a t ttn converges weakly to zero in ~ (G) and

1

(10) IIt., + t,, + ... + t , mll,=

for all subsets { t , ~ , / , , . . . . , I-m} of {1,}~=1 (.m= 1, 2, 3 . . . . ).

Suppose first t h a t G is discrete. Then let Xl, x~, x3 . . . . be any countably infinite sequence of distinct points in G, and let fn (x)= 1 or 0 as x=x,, or x r xn. For an arbitrary bounded linear functional M on ~ (G), there exists a

572

A R K I V F O R M A T E M A T I K . B d 2 n r 30 function h e ~,, (G) such t h a t M (/) = f h (x) ] (x) d x = ~" h (x) / (x) for all / s ~p (G).

G x ~ G

Since ~ I h (x)IT'< co, we have lim

M(/~)

= l i m h (x~) = 0. The equaJity (10) clearly

x 8 G n--~oo n---~r

holds for this sequence {/~}~=1.

If G is not discrete, then the H a a r measure/~ of every open set U containing the identity is positive but can be made arbitrarily small for appropriately

A ~

chosen U. I t is then apparent t h a t there exists a sequence {

,}~=x

of pair- wise disjoint measurable sets in G such t h a t /~(A~)>0 ( n = I , 2 , 3 . . . . ) a n d

1

l i r a / ~ ( A ~ ) = 0 . Write ~ (A.) as an; and define

/n(x)

as being either ~ , or 0

n--~ oo

as

xeA~

or x non

sAn.

I t is plain t h a t (10) holds for this sequence {]~}~=~.

To show t h a t ]n converges weakly to zero, consider first a n y bounded measur- able function ~0 on G. We then have

S

l n ( x ) ~ ( x ) d x _ _ < s u p [ ~ ( x ) [ . ~ , ,

I

1 X $ G

a n d thus lim f / ~ (x) ~ (x) d x = 0. For an arbitrary function h e ~p, (G) and e > 0,

n - ~ c ~ G

there exists a bounded measurable function ~ such t h a t I I ~ - h l ] p ' < s . Applying HOLDER'S inequality, we find

Ilv-hIl ,= Ilv-hIl.,< .

F r o m this, it follows t h a t l i m

f h (x)/~ (x)dx] <__ e,

and hence ]n converges weakly to zero.

L e m x n a 13. L e t G be a n y locally compact group, let q be a number > 2, and let {g~}~=l be a n y sequence of functions in ~q(G) which converges weakly to zero. Then there exist a subsequence {g,g}k~=a of {g~}~=~ and a positive con- stant A such t h a t

Hg.,+g.,+" +g.mlfo_<Am

for m = 1 , 2 , 3 . . .

This lemma has been proved for real spaces ~q b y B A N A C H and MAzuR.

(See [1], pp. 197-199.) Their proof is stated for real ~q on [0, 1] but can be carried over

verbatim

for real ~q on an absolutely arbitrary measure space. To apply the proof of BA~AcmMAzu~ to the present case, which treats a complex space ~q, we need only note t h a t a sequence

{gn}~=l

converges weakly to zero if and only if the real and imaginary parts {~g,,}.%1 and

{Yg~}n%l

converge w e a k l y t o zero with respect to bounded linear functionals on ~q which are real for real functions.

We remark also t h a t L e m m a t a A and B hold for general measure spaces.

We can now prove our Theorem. Suppose t h a t 1 < p < 2 , t h a t G is an in- finite locally compact Abelian group, and assume t h a t every function in ~p,(G*) is the Fourier transform of a function in ~p(G). The transformation Tp thus maps 2 ; ( G ) continuously onto ~p,(G*). A theorem of BA:~Acg ([1], p. 41, 573

E. HEWITr, Fourier trunsforms of the class ~v

Th6or~me 5) shows that the inverse transformation Tb -~ is also continuous. Thus there exists a constant C > 0 such that

(11)

IIT/II~, <-IIIII~ <~ CIIT/II~"

for all l e s Now consider the sequence {/=}~=1 described in Lemma A, for the space ~,(G). I t is plain that the sequence { f~}==l converges weakly to T zero in s B y Lemma B, there exist a subsequence {T/=k~k~ and a ler

positive constant A such t h a t

~ T

(12) II~x /,~ll~.<Am~.

Combining (10), (11), and (12), we see that

1 m m

(13) m~ ⁼

IIk~_l/nk

^II,-<

IIk~ Tt-k

In~-< A C m~.

1 1

As (13) holds for m = 1 , 2 , 3 . . . . , we see at once that P ~ < ~ ' which con- tradicts our hypothesis. Hence T cannot, map ~,(G) onto ~,,(G*). A theorem of BANACH ([1], p. 38, Th~or~me 3) shows t h a t Tp(~p(G)) must be of the first category; since ~,,(G*) is complete, the set of functions in s which are not Fourier transforms must be of the second category and accordingly dense.

The University of Washington, Seattle, Washington.

B I B L I O G R A P H Y

]. BAI~IACH, STEFAIff. Th6orie d e s o p 6 r a t i o n s lin6aires. M o n o g r a f j e M a t e m a t y c z n e , W a r s z a w a , 1932.

2. SEGA~J, IRVING, E . T h e class of f u n c t i o n s w h i c h are a b s o l u t e l y c o n v e r g e n t F o u r i e r t r a n s - f o r m s . A c t a Sci. M a t h . Szeged 12, 1950, p p . 157-161.

3. TITCHMARSH, E. C. I n t r o d u c t i o n to t h e t h e o r y of F o u r i e r i n t e g r a l s . O x f o r d U n i v e r s i t y P r e s s , 1937.

4. WEIL, ANDtr L ' i n t 6 g r a t i o n d a n s les g r o u p e s t o p o l o g i q u e s e t ses a p p l i c a t i o n s . A c t . Sei.

e t I n d . 869. H e r m a n n et Cie., Paris, 1940.

~. ZYCMUND, A•TONI. T r i g o n o m e t r i c a l series. M o n o g r a f j e M a t e m a t y c z n e , W a r s z a w a - L w 6 w , 1935.

Tryckt den 7 januari 1954

574

Uppsala 1954. Almqvist & Wiksells Boktryckeri AB