~eftZdFe(x), F, to(t)= On the uniqueness of minimal extrapolations

A R K I V F O R M A T E M A T I K B a n d 4 n r 3

C o m m u n i c a t e d 13 M a y 1959 b y T, NAOELL a n d LENNART CARLESOI~"

On the uniqueness of minimal extrapolations

B y Yr~CVE DOMAR

1. I n t r o d u c t i o n

L e t E be a n o n - e m p t y closed proper subset of - ~o < t < ~ and let ]0 be a complex-valued function on E such t h a t t h e class D of all functions F of b o u n d e d variation which satisfy

oo

to(t)= f r

when t E E, is non-empty. I n case there exists a function F~ E D, such t h a t V a t F~ = inf Var

F,

FeD

we call the corresponding function

/ e ( t ) =

~eftZdFe(x),

- ~ < t < ~ , a minimal extrapolation of /o-

The concept of minimal extrapolation was introduced b y Beurling [1]. Esseen [2], p. 13, pointed o u t t h a t L e m m e 1 in Beurling [1] implies t h a t a minimal extrapolation always exists if E is the closure of its interior. Actually, Beurling's methods can be used to prove more general results in this direction. We shall, however, in this paper instead t u r n to the following problem, which does n o t seem to have been treated before in the literature: If a minimal extrapolation exists, is it then unique?

I n -9 2 a n d w 3 we collect some preliminary results, a n d in w 4 we prove the uniqueness if E is a half line. The remaining cases are studied in w 5. I t turns o u t t h a t we necessarily have to lay extra conditions on /o in order to secure uniqueness. The particular case when E is the complement of a finite interval is discussed further in w 6, a n d i~ is shown t h a t we have uniqueness if f0 has t h e p r o p e r t y t h a t one (and hence every) corresponding function F has an ab- solutely continuous p a r t with a continuous derivative.

2:1 i 9

Y, DOMAR, On the uniqueness of m i n i m a l extrapolations

2. Minimal extrapolations with absolutely continuous F.

W e d e n o t e b y L~ t h e class of functions G E L 1 for which

oO

f e ux G (x) d x = 0

for e v e r y t E E, a n d b y L ~ t h e class of functions ~ E L ~ for which

r 1 6 2

f ~ (x)G (x) d x = 0

for e v e r y G E L 1. T h u s L ~ is t h e w e a k closure of t h e l i n e a r set which is s p a n n e d b y t h e functions e ~t~, t E E.

The following l e m m a is n o t e n t i r e l y new, since p a r t s of i t can be t r a c e d b a c k t o Sz. N a g y [4] a n d s e m i n a r s b y Beurling.

L e m m a . Suppose t h a t

o o

/x (t) = f e ux d F 1 (x), where F 1 E D is a b s o l u t e l y continuous.

N e c e s s a r y a n d sufficient for ]1 to be a m i n i m a l e x t r a p o l a t i o n is t h a t t h e r e exists a f u n c t i o n ~ E L ~E w i t h

[ ~ ( x ) l < l a l m o s t e v e r y w h e r e a n d such t h a t

Fi (x) ~ (x) = IF; (x) l

a l m o s t everywhere.

Proo/ o/ the necessity. I f F 1 is a b s o l u t e l y continuous a n d corresponds t o a m i n i m a l e x t r a p o l a t i o n we m u s t h a v e

m=Var F~= f tF~(x)idx< f ]F~(x)--G(x)idx,

~ 0 r - - o r

for e v e r y G E L~. H e n c e m is t h e d i s t a n c e in L 1 b e t w e e n t h e e l e m e n t F ; a n d t h e closed l i n e a r set L~, a n d b y t h e H a h n - B a n a c h t h e o r e m t h e r e exists a b o u n d e d linear f u n c t i o n a l F* w i t h n o r m ~< 1 such t h a t

while

F * ( G ) = 0 , if G E L I ( E ) , F * (_~;) = m .

T h e space of b o u n d e d l i n e a r functionals on L 1 can be i d e n t i f i e d w i t h L ~ . Hence t h e r e exists a f u n c t i o n ~ E L ~ such t h a t

I~1<1 (1)

ARKIV FOR MATEMATIK, B d 4 n r 3

almost everywhere, such t h a t

~ o

f qJ(x) G(x)dx=O,

- - o o

if

G E L~,

i.e. which belongs to

L~,

and finally such t h a t

o ~ oO

f f lF;(x)ldx,

- - r - - o c

which by (1) implies t h a t

almost everywhere.

(x) Fi (x) = I F~ (x)]

Proo[ o/ the su//iciency.

We skall use the auxiliary function K ( x ) = ~

1/ e-~'dy.

- - o 0

Let F 1 and ~ satisfy the conditions of the lemma. If F is an arbitrary func- tion in the class

D,

and if ~ denotes the ordinary convolution operation in the class of functions of bounded variation, then it is easy to see t h a t the functions

a r e absolutely continuous for every a > 0 and t h a t the derivative of their dif- ference belongs to the class L 1. Hence

o o

- - o 0

which m a y be written

Oo o 0

- - 0 o ~ o 0

T h e function K is monotonically increasing with variation 1, and hence, by a wellknown convergence theorem for Lebesgue integrals,

f dx~

21

Y . D O M A R , On the

uniqueness of minimal extrapolations

for a n y function H E L 1. Using this we see t h a t the right h a n d side of (2) converges to

as a--> + 0. On the other h a n d the left h a n d side has absolute value :Wr F o r t h a t reason

Var F 1> Var F 1

for every F E D, which shows t h a t F 1 corresponds to a minimal extrapolation.

3. On the uniqueness problem w h e n E contains a h a l f line

The difference between a n y two functions in D has a Fourier-Stieltjes trans- form which vanishes on E. If E contains a half line, i.e. an interval ( - ~o, a) or an interval (b, oo), this implies b y a theorem b y F. a n d M. Riesz [3] t h a t the difference is absolutely continuous. Hence the functions in D differ only b y their absolutely continuous parts, a n d as for questions of existence a n d unique- ness of minimal extrapolations there is no restriction in assuming t h a t /0 has the p r o p e r t y t h a t every function in D is absolutely continuous. W e shall as- sume this in the rest of this paragraph, a n d we exchange for simpUcity's sake t h e derivatives F 1, F~, F~ . . . . of functions in D to H, H 1, H~ . . . . which then are functions in a certain subclass D' of L 1.

Suppose t h a t 11 and 12 are two different minimal extrapolations of /0, cor- responding to H 1 a n d H~, respectively. The function

/:~ (t) = 89 (/1 (t) +/~ (t)) = f e '~x 89 (H1 (x) + H~ (~)) dx

- o o

is another extrapolation of /0, a n d since

f 89247 flHx(x)ldx+89 flH~(~)ldx, ⁽³⁾

it is a minimal extrapolation. W e m u s t furthermore have equality in (3), a n d this is the case only if

[H,(x)[_]H2(z)[ ⁽⁴⁾

H : (x) H 2 (x) '

for almost every x, for which t h e two members are defined. Using the function which in the sense of t h e lemma corresponds to t h e minimal extrapolation /3, (3) and (4) imply t h a t

AnKIV FOR MATEMATIK. B d 4 nr 3

and

almost everywhere. B u t

q0 (x) (H, (x) + H 2 (x)) = It/1 (x) + [/3 (x)l, (~) H, (x) = I H, (x) l,

(x) H, (x) = I//2 (x)],

f e " (//I (*) - H , (.)) d . = 0,

if t E E, and since we assume that E contains a half line, well-known properties of Fourier transforms in L 1 show that H , - H 2 coincides almost everywhere with the boundary values of a function, analytic in a half plane. Since the function is not identically zero we m a y conclude t h a t it is ~= 0 almost every- where (el. [3]). This implies that for almost every x at least one of the func- tions H i and H~ is ~: 0. We m a y conclude that

and

almost everywhere.

Apparently we have moreover

H, + H2 ~- O

(x) (H, ( x ) -

H~ (x)) = ~ (x) l H , (x) - H~ (x) ], where ~ is real-valued and

almost everywhere.

(5)

4. The uniqueness when E is a h a l f line

Theorem 1. The minimal extrapolation is unique when E is a half line.

Proof. I t is apparently enough to give the proof if E is the half line 0 ~< t < co and, by w 3, if the functions in the class D arc absolutely continuous.

We give an indirect proof, and we thus assume that there exist two different minimal extrapolations

/.

(t) = f e "z I-1. (x) d . , ~ = ], 2.

For any ~t/> 0 the function

d ~ (H, (z) - H g (~))

has a Fourier transform which vanishes if t~> 0, i.e. if t E E. The function which in the sense of the lemma corresponds to the minimal extrapolation

89

2 3

Y. DOMAR, On the uniqueness of minimal extrapolations (cf. w 3) belongs to t h e class L ~E a n d hence

f q9 (X) e tzz (H I (x) - H~ (x)) d x = 0, (6) if 2 i>0. B u t (5) shows t h a t

(H1 -- H2)

is r e a l - v a l u e d a l m o s t everywhere. B y c o n j u g a t i o n of (6) we t h e r e f o r e o b t a i n f qJ (x) e -t~* (H 1 (x) - H z (x)) d x = O,

if 2~> 0, which shows t h a t (6) holds for e v e r y r e a l - v a l u e d 2. H e n c e b y t h e uniqueness t h e o r e m for F o u r i e r t r a n s f o r m s in L 1

(x) (H 1 (x) - H 2 (x)) = 0

a l m o s t everywhere, a n d since I ~ l = 1 a l m o s t e v e r y w h e r e (w 3) we can conclude t h a t H I (X) = H 2 (X)

a l m o s t everywhere. B u t t h e m i n i m a l e x f r a p o l a t i o n s were a s s u m e d to be different, a n d hence t h i s c o n t r a d i c t i o n p r o v e s t h e uniqueness.

5. T h e e a s e w h e n E is n o t a h a l f l i n e

T h e o r e m 2. I f E is n o t a half line, t h e n t h e r e exist functions /0 which h a v e n o n - u n i q u e m i n i m a l e x t r a p o l a t i o n s .

Proo/. W e h a v e to consider two s e p a r a t e oases.

Case 1: T h e c o m p l e m e n t of E is connected. Since t h e c o m p l e m e n t is o p e n a n d is n o t a half line, i t has to be a finite i n t e r v a l . I t is a p p a r e n t l y no r e s t r i c t i o n t o a s s u m e t h a t t h e c o m p l e m e n t of E is t h e i n t e r v a l - 1 < t < 1.

Case 2. T h e c o m p l e m e n t of E is disconnected. I t is no r e s t r i c t i o n to as- sume t h a t 0 E E while t h e two i n t e r v a l s ( - a - b, - a + b) a n d ( a - b, a + b) a r e in t h e c o m p l e m e n t . (a a n d b are some p o s i t i v e n u m b e r s , b < a.)

T h e proof will be quite similar in t h e t w o cases. W e s t a r t f r o m a r e a l - v a l u e d f u n c t i o n H , which E L 1, is # 0 a l m o s t e v e r y w h e r e a n d f i n a l l y has t h e p r o p e r t y t h a t

I H (x) l e L ~ .

H (x)

B y t h e l e m m a in w 2 t h e f u n c t i o n

l ( t ) =

oO

f e ttx H (x) d :"

ARKIV FOR MATEMATIK. B d 4 nr 3 is a m i n i m a l e x t r a p o l a t i o n of t h e r e s t r i c t i o n of /o to E . T h e n let G b e a real- v a l u e d function E L~, which is ~ 0 a n d satisfies

O b v i o u s l y

a n d

[G(x)[<iH(x)l.

IH(x) l _ I H ( x ) + G(x) I H (x) H (x) + G (x)

oo

f eaX(H(x)+G(x))=f(t), if t e E .

- - O a

(7)

B y t h e l e m m a also t h i s f u n c t i o n is a m i n i m a l e x t r a p o l a t i o n , a n d b y t h e unique- ness t h e o r e m for F o u r i e r t r a n s f o r m s in L 1, t h e t w o m i n i m a l e x t r a p o l a t i o n s are different. H e n c e t h e o n l y p r o b l e m is to show t h a t we can find functions H a n d G w i t h t h e m e n t i o n e d p r o p e r t i e s .

Case 1. W e form t h e f u n c t i o n

(x) = / _ ¹ if cos x >/O, 1 if cos x < O . I t is e a s y t o show t h a t if G o EL1N L ~, t h e n

-1 2 ~r

q)(x)Go(x)dx= ~.. + --- s i n n ~ e~n~Go(x)dx.

- o o - o o

Hence, if G o E L~ N L ~

c~

f ~ (x) a o (x) d z = O, (s)

a n d since o b v i o u s l y L~ N L ~ is dense in L~, (8) holds for a n y G O E L~. F o r t h a t r e a s o n ~ E s , a n d we can choose L ~

H ( x ) 1 + x 2"

I t can b e p r o v e d b y a simple c o n t o u r i n t e g r a t i o n t h a t a n y entire f u n c t i o n of e x p o n e n t i a l t y p e 1, which belongs t o L 1 on t h e r e a l axis, h a s t h e p r o p e r t y t h a t t h e F o u r i e r t r a n s f o r m of its r e s t r i c t i o n t o t h e real axis vanishes o u t s i d e ( - 1, 1), i.e. in our t e r m i n o l o g y belongs t o L~. Hence we can choose

1 - - C O S X

G (x) = 4" x2

if t h e p o s i t i v e c o n s t a n t 2 is a s s u m e d to be so small t h a t (7) holds.

Case 2. Since 0 E E ,

oo

f G o (x) d x = 0

- o o

25

Y. DOMAR, On the uniquoness of minimal extrapolations

for e v e r y G O E L~, i.e. e v e r y c o n s t a n t f u n c t i o n belongs t o L ~~ F o r t h a t r e a s o n we can choose

H(x)

= 1 + x 2 1 The F o u r i e r t r a n s f o r m of

l - c o s b

9 c o s a x x 2

vanishes outside ( - a - b, - a + b) a n d ( a - b, a + b). H e n c e we can choose l - c o s b x

(7 (X) = ~ X2 COS a z

if t h e p o s i t i v e c o n s t a n t 2 is a s s u m e d to b e so small t h a t (7) holds.

6. A suffieient c o n d i t i o n for u n i q u e n e s s in a special c a s e

I t follows from T h e o r e m 2 t h a t in t h e cases which are c o v e r e d b y t h a t t h e o r e m t h e class of f u n c t i o n s [0 w i t h u n i q u e m i n i m a l e x t r a p o l a t i o n s is a p r o p e r subclass of t h e class of all possible f0 u n d e r consideration. T h e e x t e n t a n d p r o p - erties of t h i s subclass seem t o d e p e n d v e r y h e a v i l y on t h e a l g e b r a i c p r o p e r t i e s of t h e set E . If, for instance, E is a s u b s e t of t h e set

{an+b)~o~,

for some a # 0 a n d b, a n d if /e is a m i n i m a l e x t r a p o l a t i o n , t h e n

2 z ( t - b )

l,(t)

cos

a

is also a m i n i m a l e x t r a p o l a t i o n . H e n c e ] 0 ~ 0 is t h e o n l y f u n c t i o n w i t h a u n i q u e m i n i m a l e x t r a p o l a t i o n . On t h e o t h e r h a n d , if E is n o t of t h a t k i n d i t is e a s y t o see t h a t also t h e f u n c t i o n s

[0 (t) = c o n s t a n t h a v e u n i q u e m i n i m a l e x t r a p o l a t i o n s .

W e shall t u r n t o t h e special case w h e n E is t h e c o m p l e m e n t of a finite i n t e r v a l , a n d we shall p r o v e t h e following t h e o r e m which gives a sufficient con- d i t i o n for fo t o h a v e a u n i q u e m i n i m a l e x t r a p o l a t i o n .

Theorem 3. S u p p o s e t h a t E is t h e c o m p l e m e n t of a finite i n t e r v a l , a n d t h a t [o has t h e r e p r e s e n t a t i o n

/o (t) = f e," d Fo

(~),

where F 0 (x) has an a b s o l u t e l y c o n t i n u o u s p a r t w i t h a c o n t i n u o u s d e r i v a t i v e . T h e n i t has a u n i q u e m i n i m a l e x t r a p o l a t i o n .

Proo[.

B y w i t is e n o u g h t o g i v e t h e proof, w h e n F o ( x ) i s p u r e l y a b s o l u t e l y continuous. W e can f u r t h e r m o r e a s s u m e t h a t t h e c o m p l e m e n t is t h e i n t e r v a l

- l < t < l .

ARKIV FOli MATEMATIK, Bd 4 nr 3 The difference between a n y two functions in the class D ' (w 3) has a Fourier transform which vanishes outside - 1 < t < 1. B y the inversion theorem this has the consequence, t h a t this difference coincides almost everywhere with the values on the real axis of an entire function of exponential t y p e 1. Since b y assump- tion one of the functions in D' is continuous, we m a y assume t h a t every func- tion in the class is continuous.

L e t us therefore assume t h a t we have two different minimal extrapolations / , ( t ) = f e f t ~ t t , ( x ) d ~ , ~ = 1 , 2 ,

where H 1 a n d H~ belong to L 1 a n d are continuous. B y the above a r g u m e n t s the function

(7 (x) = H~ (x) - H~ (x)

is the restriction to the real axis of an entire function of exponential t y p e 1.

T h e function q0, which in the sense of the lemma corresponds to the minimal extrapolation

89 (/~ +/~)

(el. w 3) can be assumed to be continuous except a t certain points, where b o t h H 1 and H 2 vanish. Hence it is continuous and satisfies

except at certain of t h e zeros of (7. I t then follows from (5) t h a t the real-valued function ~ can be assumed to satisfy

everywhere, a n d it changes its sign, i.e. 1 0 ( x + O ) 4 : ~ ( x - O ) , only at certain of the zeros of the entire function (7.

L e t us first discuss the case when there are only a finite n u m b e r of changes in the sign Of ~. L e t the changes occur when

x = 2 1 . . . i~.

W e form K (x) = G (x) 1

( x - ~1) .-- ( x - ;t.)"

K is the restriction to the real axis of an entire function of exponential t y p e 1, a n d it belongs to L 1. As mentioned before (w 5) this implies t h a t it belongs t o L 1, a n d since ~0 E L ~ , we have

co 1 d x = O ,

f ~ (x) (7 (x) (x-,~) ( ~ - ;~.)

h e n c e b y (5) ^{v 2 (x) d x =} 0.

27

Y. DOYIAII., On the uniqueness of minimal extrapolations

B u t t h e i n t e g r a n d has a c o n s t a n t sign, a n d i t is n o t i d e n t i c a l l y vanishing. This c o n t r a d i c t i o n shows t h e uniqueness if t h e n u m b e r of changes in t h e sign is finite.

I f t h e r e is a n infinite n u m b e r o f changes in t h e sign of ~0, t h e r e exists a p o i n t x0, where t h e r e is no change, a n d such t h a t t h e n u m b e r of changes for x < x o a n d for x > x o is e i t h e r even or infinite. W e d e n o t e these p o i n t s b y 2~ in such a w a y t h a t

9 . . < ~ _ n < . - - < ~ _ 1 < X O < ~ 1 < - . . < ~ n < . . . .

W e can t h e n choose t h e finite open i n t e r v a l (a, b) so large t h a t t h e r e is a n even n u m b e r of changes for a < x < x o a n d for x o < x < b , s a y 2 m a n d 2 p , re- s p e c t i v e l y , a n d such t h a t

f l a ( x ) l d ~ >

+ fla(x)ld~.

(9)

~-~ - ~ b

W e f o r m t h e Junctions

a n d

K + (x) =

K _ (x) =

( x - 22) ... ( x - 22,)

(x-;t_~) ... ( x - ~-2,,) ( x - 2-1) ..- ( x - 2-~m+1)"

T h e functions are p o s i t i v e if x E (2-1, 21) a n d if x ~ (a, b), a n d i t is e a s y t o see t h a t

inf K ( x ) > sup K(x), (10)

xF.(,~_I. 21) xr b)

is fulfilled w i t h K e x c h a n g e d for e i t h e r of t h e f u n c t i o n s K + a n d K _ . H e n c e t h e f u n c t i o n

K = K + . K _ also satisfies (10).

T h e f u n c t i o n G - K belongs t o L 1 a n d is t h e r e s t r i c t i o n t o t h e r e a l axis of a n e n t i r e f u n c t i o n of e x p o n e n t i a l t y p e 1. H e n c e i t m u s t s a t i s f y

f ~ (~) a (~) K (,) d x = 0,

i.e. f ~ ( x ) G ( x ) K ( x ) d x = - + f c p ( x ) G ( x ) K ( x ) d x . (11)

a b

B u t (5) shows t h a t t h e left h a n d m e m b e r of (11) has t h e v a l u e

b

f la(x) l~p(x)K(x)dx,

a n d since ~ . K has a c o n s t a n t sign in t h e whole i n t e r v a l (a, b), we o b t a i n f r o m (11)

ARKIV FOR MATEMATIK, , B d 4 n r 3 b

flG(x)l'lK(x)ld <

a

§ fJG(x)llK(x)ldz.

- ~ b

Hence we have, using (10)

~ _ ~ - a r b

which is a c o n t r a d i c t i o n to (9). This p r o v e s t h e uniqueness in t h e general case.

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

]. A. BEURLII~O, B u r les intdgrales de Fourier absolument convergentes et leur application d u n e trans](rrmation ]~nctionnelle. Neuvi~me congr~s des math~maticiertS scandinaves, Helsing- fors, 1938.

2. C. G, ESSEE~, Fourier analysis o / d i s t r i b u t i o n / u n c t i o n s . Acta Math. 77, 1945.

3. F. and M. RiEsz, Uber die Randwerte einer analytischen F u n k t i o n . Quatri~me congr~s des math~maticiens scandinaves, Stockholm, 1916.

4. B. Sz. NAG'Z, Ober gewls~e Extremalfragen boi transforrnlerten trigonometrischen Entwick- lungen II, Bet. Math.-Phys. K1. Siichs. Akad. Wiss. Leipzig, 91, 1939.

Tryckt don 15 september 1959

Uppsala 1959. Almqvist & Wiksells Boktryckeri AB

29