A f - - ~ ~ and [~f] = y. [Ox~l 9

On the uniqueness of summable trigonometric series and integrals

TORBJ6RN HEDBERG Institut ~Iittag-Leffler

1. Introduction

The aim of the present work is to give some generalizations of a well-known theorem b y Z y g m u n d and Verblunsky, which in one of its original forms can be stated as follows [13, p. 352]:

Let ~ ~_~ a,e i€ be a trigonometric series with complex coefficients and suppose that an = o(n), n ~ ~ . I f the series is Abel summable to an everywhere finite function f E LI(T) then the given series is the Fourier series of f.

This theorem has been generalized in various directions. Verblunsky [13, p. 356]

proved it under weaker conditions on the Abel sums of the series and Shapiro [8]

has obtained analogous results in higher dimensions under the assumption t h a t

~R_<l~l<~+l la~] = o(R) as _R tends to infinity. Results have also been obtained for trigonometric integrals of one variable b y Verblunsky [12] and of several variables b y Shapiro [7]. In these theorems summation b y the Abel-Poisson or b y the Ces~ro meShod is used.

In this paper the theorem wilt be formulated in a more general form and proved for a class of summation methods which contains the Abel-Poisson method as a special case. Many of the earlier results will appear as corollaries.

The method of proof will consist in replacing summation using a given kernel H b y summation using a kernel K, which is a certain linear combination of H and its dilatations. The properties of K will be such t h a t we then can deduce our results directly.

I wish to express m y gratitude to professor Yngve Domar, who suggested the problem, for his kind interest and support.

224 ~ K I v F6~ ~ATEMATIK. Vol. 9 No. 2 2. Notations and definitions

T h e p o i n t s in t h e n-dimensional Euclid~an slo~ee I t ~ or in t h e n-dim3nsional t o r u s T ~ will be d e n o t e d b y x = (xl, x~ . . . x,). W e shall w r i t e l x l - = r =

(x~ + x~ + . . . + ~n)'~.

I f F is a n y g i v e n f u n c t i o n on R ~ a n d if R > 0 t h e n we define t h e f u n c t i o n FR b y t h e r e l a t i o n ~ R ( x ) = R"F(Rx) for all x E R%

I f a f u n c t i o n F has d e r i v a t i v e s a l m o s t e v e r y w h e r e t h e n we d e n o t e t h o s e d e r i v a t i v e s b y [OF/Oxl] etc. B y OF/Ox 1 etc. we m e a n d e r i v a t i v e s in d i s t r i b u t i o n a l sense.

W e d e n o t e t h e L a p l a c i a n b y A, i.e.

[a sl

A f - - ~ ~ and [~f] = y. [Ox~l 9

for t h e o p e n ball w i t h c e n t r e x a n d r a d i u s ~, IB(Q)I will W e w r i t e B(x, q)

d e n o t e its v o l u m e .

A r e a l - v a l u e d f u n c t i o n H o n R ~ is said to belong to t h e class 9 ( if:

(i) H is n o n - n e g a t i v e a n d t w i c e c o n t i n u o u s l y differentiable

(ii) H is radial, i.e. t h e r e exists a f u n c t i o n H 0 on [0, Jr m) such t h a t H(x) =

H0(lxl)

for all x e R ~

(iii) H is i n t e g r a b l e a n d d f i ~ , H dx = 1

i f

IB(R)I [FIdx = 0(1) as R--> oo.

~(0, R)

d e n o t e s t h e I ) i r a e measure, i.e. t h e m e a s u r e given b y t h e u~nit mass at t h e origin.

T h r o u g h o u t G will s t a n d for t h e f u n c t i o n d e f i n e d b y t h e following r e I a t i o n 1

G(x)--cn ix[2_ n if n > 3

= c z . I o g l x [ i f n = 2 where t h e c o n s t a n t s cn are chosen so t h a t A G = 8.

(iv) dHo/dr is n o n - p o s i t i v e

(v) dPtto/drP-~ O(r -(n+2+~)) as r--~ oo for some e > 0.

I n t h e sequel we shall n o t use t h e n o t a t i o n T/o, b u t write a b u s i v e l y tt(x) = H(r ) a n d H ' f o r H 0 = dHo/dr.

A locally i n t e g r a b l e f u n c t i o n _F on R ~ is said t o belong t o t h e class c ~ if:

O N T H E I ~ N I Q U E N E S S O F S U M M A B L E T R I G O N O M E T E I C S E B I E S A N D I N T E G I : ~ S 2 2 5

T h e F o u r i e r t r a n s f o r m o f a f u n c t i o n F or a m e a s u r e tt is d e n o t e d b y _F a n d respectively, b o t h w h e n t h e t r a n s f o r m is d e f i n e d in t h e o r d i n a r y sense a n d in t h e d i s t r i b u t i o n a l sense.

3. Some lemmas

B e f o r e we can p r o v e o u r m a i n t h e o r e m s we n e e d some lemmas. T h e first one is f u n d a m e n t a l for t h e m e t h o d o f p r o o f in this p a p e r a n d contains t h e m a i n n e w idea.

LEMMA 3.1. Suppose that H is a given function in the class 9g. Consider the function K defined by the relation K(x) = 2 H,(x) -~ for all x E R " ~ { 0 } . Then _K satisfies the following conditions:

(i) K is non-negative and twice continuously differentiable for x = O.

(it) K is radial.

(iii) K is integrable and f i( ~ K dx = 1.

(iv) A K = A -- M~, where A is the function defined by A(x) -= -- 2H'([xl)/lxl for x + O and where M = I R ~ A dx"

Proof. (i)--(iii) are obvious. T o p r o v e (iv) we choose a n a r b i t r a r y f u n c t i o n (p e C ~ ( l t ' ) . T h e result t h e n follows f r o m t h e following equalities:

oo

f f

( A K , q)} = ( K , Aq)} = 2 Aq:(x)dx H,(lx]) ta - -

ilr* 1

T

f f .x

= 2 lim dt Aq)(x)H(Jx]) ta --

T---> 00

1 i i n

T

f f I(x)

= 2 1 i r a q) dx A ~ dt =

T--> m

i i n 1

T

= 2 T--,limo~ ~ dx -[t \ lxl / dt---

R n 1

w i t h A a n d M

= lim f (-- q)Ar + q)A)dx = ( A -- Md, q))

T---> ao I1 n

d e f i n e d as in (iv) a b o v e .

226 ~_~KIv FSR ~ATE~_~TIK. Vol. 9 No. 2

I n o u r n e x t l e m m a we a s s u m e t h a t n > 2, b u t t h e r e s u l t also holds for n = 1 w i t h o b v i o u s m o d i f i c a t i o n s o f t h e s t a t e m e n t a n d its 10roof.

LEMMA 3.2. Let B C 1t 2 be an open ball, let G be the function defined in section 2, let H ~. 9~ and define, as in lemma 3.1, the function K by K(x) ~ 2 dt/t a for all x e I ! ~ { 0 } .

I f F E ~ satisfies the following conditions:

(i) F is u p p e r semi-continuous in B

(ii) limR..~ ( F * AKR)(x ) > f(x) for all x e B, where f is integrable, is finite in the ball B and has compact support,

then A ( F -- f , G ) > 0 in B, i.e. F - - f , G is an almost subharmonic function in B, Proof. L e t 2'0 be d e f i n e d b y F0(x ) ~ F(x) for x e B a n d zero otherwise, A trivial e s t i m a t e gives

2H'(r)

[ ~ K ] ( x ) = - -

O(r -(n+~+~))

as r - + oo a n d we h e n c e easily o b t a i n t h a t

lira (AKR * (.F -- Fo))(x ) ---- 0 for all x E B .

R-~oO

2' 0 t h e r e f o r e also satisfies (i) a n d (ii) a n d we m a y h e n c e w i t h o u t loss o f g e n e r a l i t y assume t h a t F v a n i s h e s outside B.

Assume n o w t h a t (ii) holds w i t h strict i n e q u a l i t y for all x E B a n d t h a t f is u p p e r semi-continuous. L e t /~a be d e f i n e d b y /~TR(x ) ---KR(x) for all x in s o m e ball B(O, ~) w i t h Q g r e a t e r t h a n t w o t i m e s t h e r a d i u s o f B a n d zero otherwise.

T h e n

l i m Y , A K 7 R = l i m F * A K R for all x E B .

Since f is u p p e r s e m i - c o n t i n u o u s a n d f i n i t e t h e r e exists for e a c h x o E B a n d each s > 0 a 8 > 0 such t h a t f(x)

<f(xo)

§ e w h e n e v e r ]x - - Xo] < 8. H e n c e

A F ~ 9 ( f , G)(xo) -~ (F~ , f)(x) < f(Xo) § 2e for all R larger t h a n some _R o d e p e n d i n g u p o n x 0 a n d s.

Using t h e finiteness o f

f

again we conclude t h a t

lira ((F - - f * G) * d/~R)(x) > 0 for all x E B . (3.1}

R-->o~

F u r t h e r m o r e , since

f

is b o u n d e d a b o v e , we can easily p r o v e t h a t - - f * G a n d h e n c e also F - -

f ,

G is u p p e r semi-continuous. Suppose, still on t h e a s s u m p t i o n t h a t (ii) holds w i t h strict i n e q u a l i t y , t h a t / 7 _ f , G is n o t s u b h a r m o n i c in B .

O N T I I E U N I Q U E N E S S O F S U M M A B L E T R I G O N O M E T R I C S E R I E S A I ~ D I I ~ T E G i R A L S 227

T h e n we can f i n d a h a r m o n i c f u n c t i o n u such t h a t ~b = F - - f * G ~- u has a m a x i m u m in some interior point x o of B. B u t t h e n (iv) of l e m m a 3.1 gives li--mR_~ (O 9 A/~R)(x0) <_ 0 which contradicts (3.1) since 1-TmmR~ ~ (u 9 AKR)(xo) = O.

T h e result n o w follows in t h e p a r t i c u l a r case we h a v e considered.

To prove t h e general case we choose, using t h e Vitali-Carath6odory t h e o r e m [4, p. 75], a non-decreasing sequence {fk}k%~ of u p p e r semicontinuous functions satisfying:

(i) f k - - > f a.e. a n d in L l - n o r m as /c--> oo

(ii) f , < f in B.

T h e first p a r t o f this p r o o f tells us t h a t A(F -- fk * G) ~ 0 in B. B u t fk * G converges t o f , G in t h e distributional sense a n d t h u s A ( F - - f * G) > O, which proves t h e lemma.

The following two lemmas are simple generaliza$ions of l e m m a s of Shapiro [9, p. 69--70] a n d our proofs are essentially t h e same as his.

LEMMA 3.3. Let F be an arbitrary function in 9~]i, let F be its Fourier transform and assume H E 9g.

Define the functions ~ and U by

and

2H'(Ix[)

~(x) = - - Mlx[ for x E R n ~ { 0 }

U(x) = { Co i f [x[ <- I otherwise

where the constants C a n d M are chosen such that U(O) = ~(0) = 1.

I f F is a measure such that [F[({x; N < Ix[ < 2N}) = o(1) as 25 ----> r ( F , zR - - F , UR) tends to zero uniformly in R" as R---> ~ .

then

Proof. P u t V = n - - U. Since r V E L I ( R n) we k n o w t h a t V E C I ( R n) a n d since V(0) = 0 we i m m e d i a t e l y see tha+~ If(r) = O(r) as r--~ O. :Furthermore, since H E 9~, it follows t h a t dn/dr = d/dr(-- 2 H ' ( r ) / M r ) E Li(Rn). Since d U / d r

A

is a b o u n d e d measure we can conclude t h a t V(r) = O(r -1) as r - + oo.

A

P u t / , = F . T h e n

l / (;) _P

_<

1t n

f

-< fz dl l(x) + o(1).

2N< Lx I < 2 N + 1

228 .Al~:iv ~ 6 g :~A:rE:~A'rI~:. V o l . 9 N o . 2

A

U s i n g t h e f a c t t h a t V(r) = O(r) as r - - > 0 we o b t a i n

: f (;)I <.,(x, = = o<,

^{+ .}

2N<R 2N<R

2N< 1~[ < 2N+ 1

A

F u r t h e r b e c a u s e V(r) = 0 ( r -x) as r - - ~ m it follows t h a t

alibi(x) = ~ , o = o ( 1 ) as R - + m , 2N>R 2N--< I~I<2N+ 1 2N>R

a n d t h e l e m m a is p r o v e d .

:L:EMMA 3.4. With the same assumptions and notations as in lemma 3.3 l i m s u p IF * UR(x) -- F * UR(y)I ) = O.

R-->o~ Ix--yI<R-L

T h e p r o o f is similar t o t h e p r e c e d i n g o n e a n d is t h e r e f o r e o m i t t e d '

4. The main theorems

As b e f o r e let H b e a n a r b i t r a r y f u n c t i o n in t h e class 9 ( . W r i t e f , ( x ) a n d f*(x) f o r t h e lower a n d u p p e r limits r e s p e c t i v e l y o f (F * AHR)(x ) as _R t e n d s t o

infinity,

T } t ~ o ~ M 4.1. Let F be a bounded continuous function on R n. I f (i) f . and f * are finite for all x E R ~

(ii) f . ~ Z, where Z is a locally integrable function, then f . = f * a.e. and A F = f . .

Proof. I t is sufficient to p r o v e t h a t f . = f * a.e. a n d A F = f , in a n a r b i t r a r y o p e n ball B C R ~. W e c a n w i t h o u t loss o f g e n e r a l i t y a s s u m e t h a t F v a n i s h e s o u t s i d e B, b e c a u s e i f we define a f u n c t i o n F 0 b y ~ 0 ( x ) = F ( x ) f o r x e B a n d F o = 0 otherwise, t h e n limR+ ~ ((F - - Fo) 9 AHR)(x ) = 0 for all x e B. C l e a r l y we c a n also a s s u m e t h a t Z h a s c o m p a c t suppor~ a n d we define as b e f o r e t h e f u n c t i o n s _K a n d ~ b y t h e r e l a t i o n s

K ( x ) = 2 H , ( ~ l g , x e m \ { 0 }

~(x) -

where we choose M so t h a t

f

R ~ 1

2 H ' ( l x [ )

-M lzl , . e m \ { o }

~ d x = 1.

ON T I I E SYNIQU/iil'qESS O:F S]Y]Yn~A:BLE TI:tIGONOMETI%I0 S E R I E S A N D I I ~ T E G I : t A L S 2 2 9

T h e n t h e following useful relation holds.

T

f e'

2 ( F 9 A H , ) - ~ = M ( F 9 ~: - - F , ~ r ) = .F 9 [ A K ] - - M F 9 ~ r . ( 4 . 1 ) 1

The p r o o f is a direct application o f F u b i n i ' s theorem:

T T

f

2 ( F 9 A H O ~ = 2 -~ F ( x - - y ) A H , ( y ) d y =

1 1 R n

T

/ f f

= 2 F ( x - - y ) d y A H , ( y ) -~ = M F ( x - - y ) ( x ( y ) - - z T ( y ) ) d y

R n 1 R n

where t h e last e q u a l i t y follows from t h e p r o o f o f l e m m a 3.1.

l~elation (4.1) also holds if F e ~ , which will be of use later.

I f we n o w in (4.1) let T t e n d to i n f i n i t y it follows f r o m t h e c o n t i n u i t y of F in B t h a t t h e last expression in (4.1) t e n d s to ( F , A K ) ( x ) for all x E B. A f t e r a change of scale we o b t a i n

oo

f

~1r (F 9 AH~)(x) ~ = ( f 9 ~ K , ) ( x ) f o r x r B .

R

F r o m this we conclude t h a t Z ~ f* ~ ~o, ~ ~* ~ f * in B , where ~ , (x) a n d

~*(X) are the lower a n d upper limits of F , A K R ( x ) as R t e n d s to iufinity.

W e can assume t h a t Z is finite for all x E B. L e m m a 3.2 t h e n shows t h a t t h e function r satisfies A ~ > 0 .

H e n c e A F .~ Z ~ ~u where # = A ~b is a positive measure (see e.g. [5, p. 29]) a n d hence F 9 2 H R = (/~ ~- Z) * HR.

B u t limR_~0 g * HR = Z a.e. W e also claim t h a t lim/~ 9 H R exists a.e. a n d equals some locally integrable f u n c t i o n g.

To prove this we shall use t h e wellknown f a c t [4, p. 149] t h a t

lim d#

,~o IB(r) IR( '

exists for almost all x a n d equals a locally integrable function, which we denote b y g.

:By a simple calculation we obtain

oo

R n 0 [x--yl_<r

--- f H;(r) r f d,(V)=,B(1), f

o I x - y l < , . o i~,_~.l<,.

230 ARKIV F6R I~IATE~CIATIK. Vol. 9 No. 2

Using t h e L e b e s g u e d o m i n a t e d c o n v e r g e n c e t h e o r e m we n o w easily get t h a t l i m t t 9 HR(x) = g(x) a.e.

R--> o~

F r o m this it follows t h a t f , = f * a.e. in B a n d t h a t f , a n d f * are locally i n t e g r a b l e a n d so l e m m a 3.2 gives

A ( F - - f * * G ) = 0 in B and, since B is a r b i t r a r y , t h e t h e o r e m follows.

Remark 4.1. T h e a b o v e t h e o r e m r e m a i n s t r u e e v e n if c o n d i t i o n (ii) is r e p l a c e d b y (ii') f * > X, where g is a locally integrable function.

F o r a m e t h o d o f p r o o f see [13, p. 358].

I n o u r n e x t t h e o r e m we shall r e p l a c e t h e c o n d i t i o n F c o n t i n u o u s b y a c o n d i t i o n o n t h e F o u r i e r t r a n s f o r m o f F . This will give us a generalization o f t h e results o f V e r b l u n s k y [13, p. 352] a n d Shapiro [8, T h e o r e m 1].

THEOI~,M 4.2. Let F belong to the class c)]l and let F denote the Fourier transform of F (in the distributional sense). I f

(i) 2 is a measure satisfying ]FI({x;-N < Ix I < 2N}) = o(1) as N--> ~ , (ii) f , and f * are finite for all x C R',

(iii) f , ~ Z, where g is a locally integrable function, then f , = f * a.e. and d F = f , .

Proof. L e t B G R" be an a r b i t r a r y ball. I t is sufficient t o p r o v e t h a t f , = f * a.e. a n d d F = f * in B.

L e t B o be a b o u n d e d n e i g h b o u r h o o d o f B. W e claim t h a t it suffices to consider an F w h i c h vanishes outside B 0. T o show this we choose a f u n c t i o n h E C ~ ( R ' ) w h i c h equals 1 on B a n d vanishes outside B 0. D e f i n e F 0 -~ F h . W e t h e n get Y0 = ~ * ~ w h e r e h e C~ ") a n d w h e r e

A

h(x) = 0 ( l x l - P ) as x--> ~ for all p .

(4.3)

L e t E N -~- {x E Rn; 2 N < lxl < 2 N+I} a n d d e n o t e t h e m e a s u r e _F b y #. T h e

A

f u n c t i o n F o t h e n satisfies

lFoIdx

A = o ( 1 ) a s N - ~ ~ ,

E N

(4.4)

since

OIff THE UNIQLrEI~ESS OF SUMMABLE 5rRIGOIqO~CIETRIC S E R I E S AI~D INTEGRALS 231

o<_ f <_ f f =

EN Rn EN

f f

= dl/,[ Ih(x -- y)ldx -4- d]/*l Ih(x -- y)dx q-

[yl<2 N - 1 EN 2N--1 < l y t ~ 2 N + 2 EN

f

^dl~t }h(x -- y)ldx = 11 --k h -t- I a ,

[yl_~2N+2 EN

where, using (i) and (4.3)

N--2

Ix < ]/~[(U E k ) s u p I/~(x)[ = o ( 1 ) as N---> oo

1 x _ > 2 N - - 1 N + 1

/2 _< I~I(U E~) sup I~(x)E = o(1) as _ ~

N--1 R n

I~ _< ~ (~,(Ek) sup I~(x)l)= o(1) as N - + ~o.

k=N+2 ix]>_2k--1

We can thus as claimed take F and Z to vanish outside B 0.

As we n o t e d above, relation (4.1) holds also for F E ~]~. F u r t h e r m o r e , (ii)shows t h a t 2 f l ( F , AH,) dt/t a converges to a finite limit for all x C B as T tends to infinity. F r o m (4.1) it follows t h a t limr+~ (F * Zr)(X) must exist and take finite values for all x E B . B u t we also know t h a t ( F , Zr)(X) tends to F(x) for almost all x. We can therefore assume t h a t limr+~ F * z r = F for all x and t h a t F is finite everywhere.

I f we again set ~v*(x) and q , (x) equal to the upper and lower limits respectively o f ( F * A K R ) ( x ) then we get as before z - - < f , ~ , - - < q * < f * in B.

The remaining p a r t of the p r o o f is now an a d a p t a t i o n of the arguments used b y Verblunsky and Shapiro.

Choose an everywhere finite upper semi-continuous function u _< Z.

I f q~ = F - u * G were upper semi-continuous we could as in l e m m a 3,2 conclude t h a t ~ was subharmonic.

L e t therefore D be the set o f all points in B where r is not u p p e r semi- continuous. We claim t h a t D is the e m p t y set.

Assume t h a t D is n o n - e m p t y . B y the Baire category t h e o r e m t h e r e exists a ball B ' = B ( x o, 2d) such t h a t x0 C D and such t h a t F * A H a ( x ) is uniformly b o u n d e d for x C/3 n B ' (where 13 denotes the closure of D).

F r o m this it follows b y relation (4.1) t h a t F 9 z r converges u n i f o r m l y in 13 n B ' and hence t h a t the restriction F I S n w is continuous.

232 Al~XlV 1r/51~ MATEMATIK. VO1. 9 No. 2 Choose now an arbitrary number M > [ - - u , G)(%).

Since - - u , G is upper semi-continuous and since FtgnB, is continuous we can assume t h a t d has been chosen so t h a t

qb(x) < M @ F(Xo) for all x E/7) 13 B ' . (4.5) Let x be an arbitrary point in B " - ~ B ( x o, d) not belonging to D and let x' C D 13 B ' be one of the points minimizing the distance to x from /7) gl B'.

P u t R = I X - X'[ - 1 and let U be as in lemma 3.3.

Let e > 0 be given. Since R > d -1 we can assume t h a t d has been chosen so t h a t

I F , U~(x) - - F ( x ' ) ] < I F , UR(x) - - F , UR(x')I --r (4.6) IF 9 UR(x') -- F 9 zR(x')] -~ tF * ~ ( x ' ) -- F(x')] < e .

T h a t this assumption is allowed follows from lemmas 3.3 and 3.4 and from the fact t h a t 2' 9 ~R converges uniformly to F on D 13 B'. We m a y also assume t h a t I F @ o ) - - F ( y ) [ < e for all y E D f ? B ' (4.7) and, since - - u * G is upper semi-continuous, t h a t

( - - u , G ) , UR(y) < _ M for all y E B ' . (4.8) Since ~b is upper semi-continuous in the ball B(x, Ix -- x']) we have t h a t ~b is subharmonic there and hence t h a t

+ ( x ) < r 9 u ~ ( x ) = ~' 9 V ~ ( x ) - (u 9 ~ ) 9 U . ( x ) .

(4.9)

:From (4.6)--(4.9) it follows t h a t

qb(x) < M ~ F ( x o ) - ~ - 2 e for all x e C D F I B

and together with (4.5) this gives a contradiction. We conclude t h a t r is upper semi-con~imlous in B.

Thus A ~ > 0 in B and using she same argument as in the proof of theorem 4.1 we get t h a t f , = f * a.e., t h a t f , and f * are locally integrable and t h a t A ( F - - f * * G ) = 0 in B.

This proves the theorem.

5. Application on trigonometric series and integrals

The preceding theorems can be applied to the t h e o r y of summation of trigono- metric series and integrals. In the particular case when H is the Poisson kernel t h e y will give us some of the classical results b y Zygmund, Verblunsky and Shapiro.

O N T I t E U ~ I Q U E I ~ E S S O F S U M M A B L E T R I G O N O I ~ E T I ~ I C S E R I E S A N D I N T E G R A L S 233

W e shall d e n o t e t h e p o i n t s in Z" b y m = (m I . . . . m.) a n d w r i t e 2,112 (m, x) ~1 m~xi.

Iml = (m~ + m~ + . . . + ~,,~) , =

W e s t a r t b y defining t h e s u m m a t i o n m e t h o d s in question a n d b y m a k i n g some r e m a r k s .

DE]~INITIOlV 5.1. Let H E 9g and let ~ e z " a~ e ~(~'~) be a given trigonometric series with complex coefficients.

The series is said to be summable-(H) at the point x E R" i f

R ~ o o r n C Z n

exists and is finite.

I f we h a v e t h e condition a~ = 0([m] k) for some k t h e series defines a periodic d i s t r i b u t i o n on R ~ which we d e n o t e b y f. W e can write

I n general this s h o u l d be i n t e r p r e t e d as an e q u a l i t y b e t w e e n t w o distributions. I n all eases we shall consider, h o w e v e r , t h e series on t h e left h a n d side will be a b s o l u t e l y c o n v e r g e n t for all R > 0 a n d we s i m p l y d e f i n e f 9 H R as its sum.

W e shall let f , ( x ) a n d f * ( x ) d e n o t e t h e lower a n d u p p e r limits r e s p e c t i v e l y o f f * HR(x ) as R t e n d s t o infinity.

N o w we can f o r m u l a t e some corollaries to t h e o r e m s 4.1 a n d 4.2.

THEOREM 5.1 (cf. [14]). Let ~ e z n a.~e ~(~'x) be a trigonometric series with com- plex coefficients, let H E 9g and suppose that X la.,I~(m/R) l < oo for all R > O. I f

(i) ~ # 0 - - amlml -~e~('~'~) is the _Fourier series of a continuous function F, (ii) f , and f * are finite everywhere,

(iii)

f , ~ Z for some integrab!e function g

then f , = f * a.e. and the given series is the Fourier series of f ,.

Proof. I f we suppose t h a t a 0 = 0 (as we m a y , w i t h o u t loss o f g e n e r a l i t y ) t h e n t h e conditions o f t h e o r e m 4.1 are satisfied a n d t h e result follows.

TIIEOREM 5.2 (cf. [13], 13. 356]). Let Z a , , e ~('~'*) and H be as in theorem 5.1 I f (ii) and (iii) of that theorem hold and i f in addition

(i') ~ la~] = o(N ~) as N --> oo

~'_<_ 1~[_<2N

then the conclusion of theorem 5.1 still holds.

234 ~ x z v ~'6R Z~Ar~MAr~K. Vol. 9 :No. 2

Proof. Assume t h a t a 0 ---- 0 a n d let F C L2(T ~) be t h e function whose F o u r i e r series is

~,,,:o-

laml ]mJ 2ei(m'~). Consider _F as a t e m p e r e d distribution on R ~.

I t s F o u r i e r t r a n s f o r m consists of point masses a t t h e points {2~m}. Condition (i') implies t h a t this measure satisfies condition (i) of t h e o r e m 4.2 f r o m which t h e t h e o r e m follows.

N e x t we shall prove a result concerning trigonometric integrals.

THEOREM 5.3. Let ] be a given function in the Schwartz class 3' of tempered distributions and let H C c2g be such that f H R E L I ( l t ") for all R > O.

Write = fo + fl, where fo has compact support and where f l vanishes in a neighbourhood of the origin.

Let f e ~5"' be the inverse _Fourier transform of f and let f , ( x ) and f*(x) denote the lower and upper limits of f , HR(x ) as R tends to infinity.

If

(i) -- lyl-2 f(y) is the _Fourier transform of a continuous and bounded function _F1 (ii) f , and f * are finite for all x q t l n

(iii) f , > Z, where Z is a locally integrable function, then f , = f * = f a.e.

Proof. L e t fi be the inverse F o u r i e r t r a n s f o r m of ] i , i ---- 0, 1. T h e n fo E C a (lt") a n d hence l i m f o , H R = f 0 for all x. ~'rom t h e o r e m 4.1 i t follows t h a t A F 1 = f l a n d t h e t h e o r e m follows.

Remark 5.1 (el. [7]). Condition (i) is satisfied if for instance (a) ](y)(] -~-lyl2) -1 e L 1

or if

(b) f(y)(1 + ly/2) -~ e L 2 a n d f ] ( y ) ( 1 -6

lf]12)-lei(x'Y)df]

B(o, R)

converges poin~wise to a continuous function as R t e n d s to infinity.

B y t h e same m e t h o d , b u t using r 4.2 instead of t h e o r e m 4.1, we o b t a i n the following result.

THeOReM 5.4. Let f, f, f , , f * and H be as in theorem 5.3.

I f (ii) and (iii) of theorem 5.3 hold and i f in addition

f A

(i')

I fldy N<IyI<_2N

= o ( N z) as N---> oo

then f , = f * = f a.e.

O N T t I E U N I Q U E N E S S O F S U M M A B L E T R I G O N O M E T R I C S E R I E S A N D I NT EGI ~ ,/ kL S 235

6. Exceptional sets

I n t h e p r e c e d i n g sections we h a v e t h r o u g h o u t a s s u m e d t h a t f , H~ is b o u n d e d as a f u n c t i o n o f R at all p o i n t s x. This condition can be s o m e w h a t w e a k e n e d a n d results c o r r e s p o n d i n g to t h o s e o b t a i n e d b y V e r b l u n s k y [13, p. 356] a n d S h a p i r o [8, t h e o r e m 2], [7] c a n be p r o v e d .

I f n ---- 1 it is for i n s t a n c e sufficient b o t h in t h e o r e m s 4.1 a n d 4.2 t o assume t h a t f , a n d f * are finite e x c e p t in a d e n u m e r a b l e set E if in a d d i t i o n we k n o w t h a t

f * H R ( x ) ~ - o ( R ) as R - + oo for all x C E .

I f n ~_ 2 it is sufficient in t h e o r e m 4.1 t o assume t h a t f , a n d f * are finite e x c e p t in a set o f zero c a p a c i t y w i t h r e s p e c t t o t h e kernel - - G. I n t h e o r e m 4.2 it is sufficient t o assume for n ~ 2 t h a t f , a n d f * are finite e x c e p t in a set w i t h o u t finite cluster points.

T h e following r e s u l t seems to be new. F o r t h e sake o f simplicity we do n o t f o r m u l a t e it in full generality.

Ta~OREM 6.1. Suppose n ~ 2. Let ~ e z n a.~ e i(m'x) be a given trigonometric series with complex coefficients and assume that H E 9~ is such that X ]amH(m/R)] <

for all R > O. Assume further that E is a bounded closed set of capacity zero with respect to the kernel -- G.

if

(i) [a l=o(N2)

N ~ i m I < _ 2 N

(it) l i m ~ + + a , ~ H ( m / R ) e i(m'x) ~ - C for x ~ E , where C is a constant,

(iii) there exists ~ ~ 0 such that X a.,H(m/R)e i('~'~) ~ O(R 2-~) as J~--> ~ for x E E ,

then a.~-~ O, m :/: O, and a o ~- C.

Proof. W e m a y w i t h o u t loss o f g e n e r a l i t y assume t h a t a 0 = 0.

L e t F o E L2(T ~) be t h e f u n c t i o n w h o s e F o u r i e r series is 2 7 - a,~lm1-2 e i('~'~).

Consider F o as a periodic f u n c t i o n o n R n a n d let B C R ~ be a n a r b i t r a r y open ball c o n t a i n i n g E . As in t h e p r o o f o f t h e o r e m 4.2, we can m u l t i p l y _F 0 b y a f u n c t i o n h E C~~ n) w h i c h equals 1 on B a n d w h i c h vanishes outside a neigh- b o u r h o o d B o o f B. I t is sufficient t o p r o v e t h a t t h e f u n c t i o n F ---- h F o equals C a.e. on B. P u t Z = Ch a n d let as before u d e n o t e t h e f u n c t i o n - - 2 H ' ( r ) / M r .

F r o m (it) a n d (iii) it follows t h a t

oo

f (F * AH,)(x)dt/t 3

1

236 AI~KIV FOB, ~ATEMATIK. Vol. 9 :No. 2

converges b o t h w h e n x r a n d w h e n x C B ~ C E a n d hence, using (4.1), t h a t F * ~v(X) t e n d s t o a finite limit for all x E B as T t e n d s t o i n f i n i t y .

W e can t h e r e f o r e assume t h a t _~ has b e e n chosen finite e v e r y w h e r e a n d such t h a t l i m T _ ~ | F for all x E B .

L e t D be t h e set o f p o i n t s w h e r e 2' is n o t c o n t i n u o u s a n d suppose t h a t D is n o n - e m p t y . Using t h e B a i r e c a t e g o r y t h e o r e m we t h e n can p r o v e t h a t t h e r e exists a bM1 B ' = B(xo, 2d) such t h a t its c e n t r e x 0 belongs t o D a n d such t h a t R~-2(F * ~ H ~ ) ( x ) is u n i f o r m l y b o u n d e d for x E D f3 B ' a n d for all R > 1. B y (4.1) this implies t h a t F * Zv converges u n i f o r m l y in /~ t3 B ' a n d hence t h a t 2'[~ n 4, is continuous.

W r i t e q~ = F - - ;~ 9 G a n d n o t e t h a t X * G is continuous. P r o c e e d i n g as in t h e p r o o f o f t h e o r e m 4.2 we g e t t h a t r is c o n t i n u o u s in a n e i g h b o u r h o o d o f Xo a n d t h u s t h a t F is c o n t i n u o u s in B.

Using (4.1) once m o r e we s e e t h a t lim ( F 9 AKR)(x) = C for all x E B n C E a n d we can conclude f r o m l e m m a 3.2 t h a t t h e s u p p o r t o f d ~ is c o n t a i n e d in E . Since E has c a p a c i t y zero a n d since ~ is c o n t i n u o u s in B it follows f r o m a classical t h e o r e m [3, t h e o r e m V I I . I ] t h a t q~ is h a r m o n i c in t h e whole o f B, which gives t h e t h e o r e m .

7. A pointwise saturation theorem

I n this section we shall consider a p r o b l e m o f a slightly d i f f e r e n t t y p e .

I t is a w e l l - k n o w n t h e o r e m b y I-~ille t h a t if t h e Abel-Poisson m e a n s u(r, x) o f a f u n c t i o n f E C(T) satisfy

u(r, x) - - f ( x )

lim ---- 0 (7.1)

r-+l--0 1 - - r

u n i f o r m l y in T t h e n f equals a c o n s t a n t [3, p. 122]. F o r an a c c o u n t o f f u r t h e r results in this d i r e c t i o n see e.g. S u n o u c h i [10].

W e shall here p r o v e a t h e o r e m which shows t h a t t h e a b o v e result r e m a i n s t r u e e v e n u n d e r t h e m u c h w e a k e r a s s u m p t i o n t h a t (7.1) holds pointwise. A n d r i e n k o [1]

has r e c e n t l y o b t a i n e d a similar result for (C,1)-summability.

L e t N b e t h e d i s t r i b u t i o n on R whose F o u r i e r t r a n s f o r m is ~ ( t ) = - - i 9 sign t, define 97 = f 9 N (where t h e c o n v o l u t i o n is d e f i n e d b y m e a n s o f t h e F o u r i e r t r a n s -

n + i

form) a n d let P be t h e Poisson kernel, i.e. P ( x ) = cn(1 + lx[~)--~ - w h e r e t h e c o n s t a n t s cn are chosen so t h a t /3(0) = 1.

T h e m a i n results o f this section is t h e following t h e o r e m .

Ttt:EORE~I 7.1. Let n = 1 a n d let X be the space L~(R) or LI(T). S u p p o s e that f E X is f i n i t e everywhere and let - - ~ < a < b < + ~). I f

O N T I - I E U N I Q U E L V E S S O:F S U S ~ I V [ A B L E T I : t I G O N O l V I E T ~ I C S E R I E S A N D I N T E G R A L S 237

lim JR(f , PR(x) -- f(x)) = g(x) for all x E [a, b] , (7.2)

R--~ oO

where g is a locally integrable and finite function, then

dx -- -- g in [a, b] .

Proof. We start by noting t h a t the proof of theorem 4.2 actually gives a stronger result t h a n stated in t h a t theorem. We have, in fact, only used the assumptions (ii) and (iii) to prove t h a t the integral

co

(f 9 H,/(x/g

1

is convergent and t h a t

and

f dt

~*(x) : lim 2R 2 ( f 9 H,)(x) -~

R - + oo R

09

f ( f dt

~,(x) = lim 2R 2 9 H,)(x) t~

R - + ~ R

both are finite for all x and t h a t ~ , >_ Z for some locally integrable function Z.

The assumptions on f , and f * in theorem 5.2 and 5.4 could therefore be replaced by, for instance, t h e assumption t h a t for all x E [a, b]

and

( f , H , ) ( x ) ~ <

1

f d,

lim 2R 2 ( f * Ht)(x) -~ : g(x),

R

where g is a finite and locally integl'able function.

Using this observation it is now an easy m a t t e r to deduce theorem 7.1.

I f we set y = . R -1 we can write

Y

l f d

R ( f 9 PR -- f ) = y dt ( f * P,-Jdt

0

238 ARKIV FOR MATEMATIK. Vol. 9 No. 2

L e t ~0 be t h e t e m p e r e d distribution whose Fourier t r a n s f o r m is T h e n t h e Fourier t r a n s f o r m of d/dt(f. P,-1) equals

d ^^ d ] ^ ^ P

d--t ( f P ~ ) = ~ ( " e - ' l Y l ) = q)" P ' - ' = ( ~ v * t-,) 9 Using this we can write (7.2) as

l/

Y

lim - (q *P,-,)(z)dt=g(x) y~o+ Y

0

which, after a change of variable, gives

-- IY[f(Y).

ix)

f

lim .R (~v 9 P,)(x) -~ = g(x).

R--->- oo R

B y m e a n s of a p a r t i a l i n t e g r a t i o n we finally obtain

oo

f

lim 2R ~ (~0 9 Pt)(x)~ = g(x)

R.--> oo R

for x E [ a , b ] . B u t , since f E X, we k n o w t h a t

2 N

f ^l~ldx

= o(N2) (or ~ I~(k)l = 0(5"2))

N

and hence we can b y t h e observation in t h e beginning of this proof, use t h e o r e m 5.2 or 5.4 to conclude t h a t

On t h e other h a n d ,

a n d t h u s

= g a.e. in [a,b].

(f-)" (t) = - ~ f~t) sign t

,f(t/sign

\dx/ (t)=- t - - - which proves t h e theorem.

_Remark 7.1. I t m i g h t be w o r t h pointing o u t t h a t t h e particular p r o p e r t y of t h e Poisson kernel t h a t m a k e s t h e above p r o o f work is t h a t t h e function (-- x P ) - belongs to c~..

B y an analogous m e t h o d we can, using t h e o r e m 5.3, o b t a i n t h e following result.

O1% r T H E U P q I Q U ] g N E S S O F S U ] ~ M A B L E T R I G O N O M E T I ~ I C S E R I E S A N D I N T E G R A L S 239

T H E O R E M 7.2. L e t n ~ 2 a n d let B c R n be a n arbitrary open ball. A s s u m e that f E L ~ ( R ~) N LI(R ") is f i n i t e - v a l u e d . I f

lim R ( f 9 PR(x) - - f ( x ) ) = 0 f o r all x e B ,

R - + vo

then f = O i n B .

As a corollary to theorem 7.1 we can get the result obtained by Andrienko [1]

in the case when g = 0 .

COROLLARY 7.1. L e t X , f a n d g be as i n theorem 7.1 a n d denote the F e j & kernel

by n. If

lim R ( f * DR(x) ,-- f ( x ) ) = g(x) f o r all x C [a, b] (7.3)

R.--> oO

then d ~ d x = - - g i n [a, b].

Proof. I t is sufficient to prove t h a t (7.3) implies (7.2).

Fix x E R and assume t h a t f ( x ) = 0 which can be done without loss of generality. Set ~o(t) = ~(t)e ~x' i k f ( - - t)e-~%

We can t h e n write

oo oo oo

i f f( i f

f * P R ( x ) ~- ~ t ) e - ~' e ~x' dt = - ~ y~(t)dt u - - e-U d u =

- - m 0 t ] R

oO u R

i f f

- - 2~ ue - u d u yJ(t) 1 - - dt = ue - u Q ( u R ) d u ,

0 0 0

where, by assumption, Q(u) = g(x) 9 u -z ~- o(u -z) as u - + oo and where IQ/ is bomlded. B y means of a simple change of variable we now immediately obtain our corollary.

We finish this section b y remarking t h a t lemma 3.2 could be formulated as a pointwise saturation theorem. The method of proof of the lemma does in fact give the following result, which is similar to ~ theorem proved by H. S. Shapiro [6, p. 27]

(in the case when n = 1) under stronger assumptions on f.

T H E O R E M 7.3. S u p p o s e that H is a positive radial f u n c t i o n on R n w h i c h satisfies

(i) f H dx = 1,

R n

(ii) f . dx =

^{a s}

[~[ > _ R

240 ARKIV FOR MATEMATIK. Vol. 9 ~qo. 2 I f f is a bounded continuous function on R ~ for which

lim R2(f 9 HR(x) -- f(x)) = g(x)

R - + oo

at each point x C R ~, where g is finite and locally integrable, then

zf=g.

(7.4)

Proof. F r o m condition (ii) it follows t h a t it is sufficient to prove the theorem in the ease when f (and hence also g) has compact support.

B y solving the equation

a2K n - 1 OK A K - - Or 2 ~- r ar -- H we get a function

K ( x ) = / r l " " d r f H ( y ) d y

lxl Ixl >- r

satisfying A K = H - (~ and (7.4) can be written lim ( f 9 AKR)(x) = g(x) .

R - + oo

The argument used in the proof of lemma 3.2 now gives the result.

Note. After the completion of the manuscript the author was informed t h a t prof. H. Berens recently has obtained theorem 7.1 b y a different but related argument [2].

ON T H E U N I Q U E N E S S OF S U M M A B L E T R I G O N O M E T R I C S E R I E S AI~D I N T E G R A L S 241

References

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2. BEREI'~S, H., On the approximation of Fourier series b y Abel means. (To appear.) 3. C~LESON, L., Selected problems on exceptional sets. V a n Nostrand, Princeton 1967. (Mimeo-

graphed, Uppsala 1961.)

4. S ~ s , S., Theory of the integral. Monografie matematyczne, Warsaw 1937.

5. SCHWARTZ, L., Thgorie des distributions. H e r m a n n , Paris 1950.

6. SHArIRO, H. S., Smoothing and approximation of functions. V a n l~ostrand, New York 1969.

7. SHAPIRO, V. L., Cantor-type uniqueness of multiple trigonometric integrals. Pae. J . Math. 5 (1955) 607--622.

8. --~>-- Uniqueness of multiple trigonometric series. A n n . Math. 66 (1957), 467--480.

9. --~>-- Fourier series in several variables. Bull. Amer. Math. Soc. 70 (1964), 48--93.

I0. Su~oucHi, G., Saturation in the theory of best approximation. On approximation theory.

Proceedings o] a Con]erence at Oberwolfach, edited b y Butzer a n d Korevaar. Birk- h~uscr, Basel 1964.

11. VERBLUNSKY, S., On the theory of trigonometric series. I I . Proe. London Math. Soc. 34 (1932), 457--491.

12. --,>-- On trigonometric integrals a n d harmonic functions. Proc. London Math. Soc. 38 (1935), 1.--48.

13. ZYGMUND, A., Trigonometric series. I. Cambridge U n i v e r s i t y Press, Cambridge 1959.

14. --~>-- Sur les s@ries trigonomgtriques sommables par le proc~d@ de Poisson. Math. Z. 25 (1926), 274--290.

Received 11.2.1971. TorbjSrn Hedberg

HSgskolenheten S-95106 Lule~

Sweden