Uniqueness of square convergent trigonometric series

UNIQUENESS OF SQUARE

CONVERGENT TRIGONOMETRIC SERlES

YOUNG-HWA HA* AND JIN LEE

§l. Introduction and the main theorem

It is well known that every periodic functionf EL'([O, 21r)), P> 1, can be represented by a convergent trigonometric series called the Fburier series off. Uniqueness of the representing series is very im- portant, and we knOw that the Fourler series of a periodic function

f E V([O,21r))is unique.

More- general functions and even distributions can be represented by trigonometric series. The uniqueness problem in this case was first answered by Georg Cantor. He proved that if a trigonometric series converges to 0 everywhere, then all the coe1fcients are o. ^{But, the}

uniqueness problem for multiple trogonometric serieshas rather differ- ent aspects. For multiple series there are more than one summation or- dering. Examples of usually considered summations for 2-dimensional

. ^{~oo ~oo}

senes L..m=-ooL..n=-oo am,n are

1) iterated: E:=_oo(E~=_ooam,n), 2) circular: limr_ooEm2+n2<r2am,n,

3) rectangular: limp,9-ooEl:l~p E,n'~9^am,n,

4) square: limr_ooElml~r Elnl~ram,n.

H f is a periodic function of n variables in V(Tn), p > 1, then the Fourier series off is square convergent to f a.e.. But, rectangular convergence and circular convergence are not guaranteed. 50 square convergenceseemsto be a more natural convergence mode(cf. [F), [5],

and [TJ).

Received October 19, 1994.

1991 AMS Subject Cll!ISSification: 42A20.

Key words: trigonometric series, Fourier series, uniqueness of trigonometric series.

*Partially supported by a grant from Ajou University, 1994.

For the uniqueness of multiple trigonometric series the following results have been proved :

i) Uniqueness for iterated convergence is an immediate conse- quence of the Cantor's theorem.

ii) Ifa double trigonometric series converges circularly to 0 ev- erywhere, then all the .coefficients are 0 (cf. [C]).

iii) Ifa multiple trigonometric series converges rect.angularly to 0 everywhere, then all the coefficients are 0 (cf. [AWl}, [AW2] , and [AFR]).

But, there are no known results about the uniqueness for square con-

overgence. In this paper we give a partial answer to the uniqueness problem for square convergence.

A d-dimensional mUltiple trigonometric series is a series of the form

. .L

kEZd

with x ERd. For every nonnegative integer n we let

£n = {k⁼ (k1 , · · · ,kd)^E zdl ^{max Ikjl} ~n},

1~j~d

and

ifn=O

ifn~l.

Then

1) £0 C £1 C £2 C '.',

2) £n = Uk=oBk ,for every n~ O.

A subset E ofZd is called a (d-dimensional) squarelike indexing set if

£n-1 C E C £n for some positive integer n. If£n-1 C E C £n and E =J £n-1' then the number n is called the order ofE and denoted by

IEI. We also consider£0 as a squarelike indexing set of order O. Given multiple series S =^{EkEZd ak,} a squarelike partial sum of S is defined to be SE = EkEEakfor some squarelike indexing set E. We say that

EkEZd ak converges squarelikely to a provided that for every e > 0 there existsN such that ifEis a squarelike indexing set and IEI ~N, then ISE - al < e. A partial answer to the uniqueness problem for square convergence is now given by the following theorem, which is the main theorem in this paper.

THEOREM. Suppose that a 2-dimensional trigonometric series S(x,y) =

L

m,nEZ

converges squarelikely to 0 everywhere and its squarelike partial sums Se(x,y) satisfy SE(X, y) = o(IEI-¹) as IEI ^-+ 00. Then all the coefIi- cients am,n are O.

In fact, we can prove a similar result for higher dimensional trigono- metric series. For that we need the condition that the squarelike partial sums SE(X) are ofo(IEj-d+l) as IEI ^{-+ 00,} where d is the dimension.

The proof for this case is quite complicated; while the result is not more satisfactory. So we consider just the 2-dimensional case.

§2. Second formal integral and Schwarz connectors The second formal integral ofa double trigonometric series

'"" a eimx einy

~ m,n m,nEZ

is a 2-dimensional series

L

m,nEZ

where

(1)

1 1

am,n (im)2 (in)2' if m::j:. 0 and n ::j:. 0,

am,n(!x2)(i~)2' ifm=Oandn::j:.O,

1 1 ₂

a^m,n(imV(2"Y)' if m::j:. 0 and n = 0, am,n( !x 2)( !y2), if m = n = O.

PROPOSITION 1. Let S(x,y) = Em,nEZam,neimzeiny.

H S = Em,nEZam,n converges squarelike1y, then the second formal integral of S(x,y) converges absolutely and uniformly to a continuous function.

Proof. By the hypothesis, the squaxelike partial sums ofS axeboun- ded. ChooseM >0 such that ISEl $ M/2 for every squaxelike indexing set E in Z2. Since

ifIml ~ 1ⁿl,

ifIml < 1nl, we have lam,nl $ M for all m, n E Z. Hence

< 00.

The assertion now follows. 0

We now define a well-ordering-< onZ2for which the immediate suc- cessor and predecessor of(m,n) axedenoted by (m,n)+ and (m,n)-, respectively. Consider (0,0) as the smallest element, and for each (m,n) E Z2 define

(2) (m,n)+ =

(0, -1), (m+l,n), (m,n+l), (m-l,n), (m,n-l),

if m = n = 0,

if n < 0 and n <m < -n, ifm >0 and -m$ n < m, if n >0 and -n<m $ n, if m <0 and m $ n $ -rn.

We thus have an ordering on Z2 so that

(3) (0,0)-< (0, -1) -<(1,-1)-< (1,0)-<(1,1) -<(0,1)-< (-1,1)-<(-1,0)

-<(-1, -1) -<(-1, -2)-< (0, -2)-<(1, -2) -<(2, -2)-<(2, -1) -< ....

One can imagine this sequence as an outward spiral which traces Z2 counterclockwise with starting point (0,0). Note that this sequence fills up B_n successively. Hence, we have

i) LN = {(m,n) EZ21(m,n) ::S(-N, -N)} for N _~ 0, (4) ii) BN = {(m,n) E Z21(-N +1,-N) ::S (m,n) ::S^{(-N, -N)}}

for N > ^0,

and for each(p, q) E Z2 the sum over the indexing set {(rn,n)I(m,n) ::S (p, q)} defines a .'Jquarelike partial sum.

PROPOSITION 2. Let A(p,q) = L:(m,n)-«p,q) a(m,n) and D.b(p,q) =

b(p,q) - b(p,q)+ for (p, q) E Z2. Then for N ; 0 i)

L

L

(m,n)E8N (m,n)E8N

+A(-N,-N)b(-N,-N)+ - A(-N+l,-N+l)b(-N,-N)+,

andfor N _~

°

ii) ~L....J a(m,n) (m,n)b

(m,n)ECN

=

L L

O$k$N (m,n)E8k

Proof. From the definition of A(m,n) we have

L

(m,n)E8N

L

=

L

L

(m,n)E8N (m,n)E8N

From BN = {(m,n) E Z21(-N +^1,-N) ::S ^(m,n) ::S ^(-N,-N)} for

N > 0, we have

L

L

(rn,n)EBN (-N+l,-N)~(m,n)~(-N,-N)

L

(-N+l,-N+l)~(m,n)~(-N,-N+l),

- L

(m,n)EBN

Substituting the last expression into the above equation, we obtain i).

Since a(O,O) = A(o,o), i) then implies

L a(m,n)b(m,n)= L L a(m,n)b(m,n) +A(O,o)b(o,o)

(m,n)E£N 15"'5^N (m,n)EBA;

= L ( L A(m,n)~b(m,n)⁺A(-k,-k)bC_k,_k)+

15k5N Cm,n)ES"

- AC-k+l,-k+l)bC_ k+ 1,_k+1)+)+Aco,o)b(o,o)

= L L ACm,n)~bCm,n)

15k5N(m,n)ES"

+ [ L (A( -k,-k)bC-k,-k)+ - A( -k+l,_k+1)b( -k+l,-k+I)+ ) +A(O,O)bCO,O)].

15 k 5 N

The expression in the bracket is equal toA(-N,-N)b( -N,-N)+. We thus obtain ii) and completes the proof. 0

For j = 1,2 let ej be the unit vectorin lR? whose j-th coordinate is 1, and for j = 1,2, hE lR, and x E lR^2, let

~J,hf(x)= f(x) - 2f(x+hej)+f(x +2hej), and

-~},hf(x) ~1(x- hej) - 2f(x) +f(x+hej).

The Schwarz difference :6.(ht

,k/(x) and difference qU:otient Q(ht,k/(X) is defined by

t st ~(h\)f(x)

~(h,k)f(x) ⁼ ~i,h~~,kf(x) ^and Q(h,k/(x)= (h₂)s(k2)t'

for (8,t) E {O,l}2, (h,k) E]R2 - {(O,O)}, and x E1R2. We now define the Schwarz connector Ds,t f(x) by

Ds,t f(x) = lim Q(sht k)f(x),

h,k-O '

where the limit is taken over such (h, k) E 1R2 - {(O, On as 1/2 ::;

Ih/kl ::;^2.

PROPOSITION 3. Suppose that the second formal integral of the doubletrigonometric seriesI:m,nEZam,neimxeiny converges everywhere toa function f(x, y). Then for each (8,t) E{O,IF the Schwarz differ- ence quotient off(x,y) is given by

(5) Q(ht,k)f(x,y) =

L

m,nEZ

where

~".(r)

kr/2 ' -r

1, if k= 0,

if m = n = O.

ifm =1=0,n = 0, ifm = 0,n =1= 0,

~s,h(eimx) ~t,kOy2) am,n(im)2(h2)S (k 2)t '

~s,h(i^{x2 )}~t,k(i^y2)

am,n (h 2)s (k2)t'

= (6)

and Ao,k(r) = Al,k(r)r²e^ikr.

Proof. Since f(x, y) = I:m,nEZa~,n(x,y)eimxeiny, wherea~,n(x,y) are given by (1), we have

Q(h\/(x,y) =

L

m,nEZ

But, by the definition of the Schwarz difference, we have Q^ll,t (a* (x y)eimxeiny)

(h,k) m,n ,

~s,h(eimx )_~t,k(einy)

am,n (im)2(h 2)s (in)2(k2)t' ^{ifm =1= 0,}n =1= 0,

~s,h(i^{x2 )}~t,k(einy) am,n (h 2)s (in)2(k 2)t'

It is easy to check that

(7)

{

eijwAl^)-(r), ifp= 1, .6.p,r(^eijw) = '

( . ·)2(_{ZJ r}^2)p _{ijw \ _() if 0} e "'0,) r , p = ,

for j #^{0, and}

(8)

ifp= 1,

ifp = o.

and completes the proof. 0

LEMMA 4. LetPm(h) = sin2(mh/2)/(mh/2)2 and.6.p.m(h) = Pm+!

(h) - Pm(h). Then, there are positive constants Cl and C2 such that if0< Ihl <1, then

(9)

L

Iml~N

for every N with 1~ N ~ l/lhl, and

(10)

L

Iml~N

forevery positive integer N.

Proof. Let us define 9(t) =(sin(t/2))/(t/2) ift #0, and 9(0) =1.

Itis easy to check that 19(t)1 ~ 1and 19(t)1 ~ 2/ltl for everyt. We can also find two constantsCa and C₄ such that

Since

D../-lm(h) = (O((m +l)h))2 - (O(mh))2

= (O((m +l)h) +O(mh))(O((m+l)h) - 8(mh)), we have

j

ID../-lm(h)I~ ^2(O((m+l)h) - O(mh)) =21 8'(t)dtl·

mh

Hence, if 0< lhl < 1 and 1S^N ~ I/lhl, then

tm+l)h

L

L li"

Iml~N Iml~N ^mh

j

~ 2 18'(t)ldt

-(N+I)lhl

f2N1hl

~ 4C³io ^{tdt = 8C3N2h2,}

and so we obtain (9) with Cl =8C3 . In particular, L:lml~rhID../-lm(h)I

S Cl. So in order to obtain (10), it suffices to showL:1ml>rh 1D../-lm(h)1 S C for some constant C. But, if Iml > Ikl' then

2 2 6 IO((m +^l)h) +^O(mh)1 ~ I(m +^l)hl + Imhl ~ Imhl

and

2C4

18((m+l)h) - 8(mh)1 _~ sup IO'(t)I·lhl ~ -I-I'

tE[mh,(m+l)h] m

Hence, if 0< Ihl < 1, then

"_L..J If),._/-lm(h)1 <_- 24C_Ihl⁴ "_L..J _1_{m2 -}< 24C_Ihl⁴ (h² ₊

foo

1 t 2

Iml>rh m>rh TiiT

= 24C4(lhl + 1) <= 48C4 .

The proof is now completed. 0

LEMMA 5. For each (s, t) E {O,IF and rn, nE Z, let>'(~,n)(h,k) =

>'s,meh)>'t,n(k) and~>'(~,n)(h,k) = >'(~,n)(h,k) - >'(~,n)+(h,k), where

>'s,m and>'t,n are definedasin Proposition 3. Then there isaconstant C >0 such that

(11)

L

L

N?l (m,n)EBN

for all(h, k) E]R2 with 0 < Ihl < 1,0< Ikl < 1, and 1/2 ~ Ih/kl ~ 2.

Proof· Considerl:(m,n)EBNI~>'(~,n)(h,k)/. By the definition ofthe ordering -<, we can write

L

N-l

L

N-l

+

L

N

+

L

m=-N+l

N

+

L

=(

L

m=-N+l

N-l

+ (

L

I)

N

+ (

L

N

+ (

L

m=-N

Since IAu,-N(r)1 =^IAu,N(r)1 for u =0,1, we have

~ I~A(~,n)(h,k)l::;

2(

(m,n)EBN Iml$2N

+

2(

Inl:52 N

Let Fs,t(h, k) = L:N~l L:₁m!:52 NIAs,m+l(h) - As,m(h)IIAt,N(k)l. Then, (12) ~ ~

L

N~l (m,n)EBN

Now suppose 0< Ihl< 1,0< Ikl < 1, and 1/2::; Ih/kl ::; 2. Then, IAo,N(k)1 ::; IAl,N(k)1 =JlN(k),

and

IAl,m+l(h) - Al,m(h)1 = I~Jlm(h)l,

where I-lN(k) and _~I-lm(h)are defined as in Lemma 4. So we have Fl,t(h, k) ::;

L

L

N~l Iml$2N

Hence, from II-lN(k)1 ::; 1and (9) it follows that

L

L

L

l$N$~ Iml$2N l$N$~

On the other hand, from II-lN(k)1 ::; Nt_{k 2 ::;} N\~2 and (10) it follows that

We thus have Fl,t(h, k) ::; C for some constant C. By the same argu- ment we also have F1,S(k, h) ::; C.

Now consider PO,t(h, k) and PO,S(k, h). Since IAo,m(h)1 < h² and Ao,o(h)=0,

L

and so

Ot ) ' " 1 2 ' " 1 2 4

P , (h, k _~ L- N . 8Nh ·IILN(k)1 _~ L- N ·8Nh . N₂k₂

N?l N?

=128

L

N?l

for some constant C, and by a similar argument pO,s(k, h) ::; C.

Putting these result together into (12), we can now derive (11). 0 The following proposition is the main result in this section.

PROPOSITION 6. Suppose that a 2-dimensional trigonomeric series S(x, y) =

L

m,nEZ

converges squarelikely to 0 everywhere, and that all of its squarelike partial sums 8E(X,y) satisfy 8E(X,y) = o(IEI-¹) as IEI ^{-+ 00.} Let f(x, y) be the second formal integral of 8(x, y). Then, for each (s,t) E {0,1}2 the Schwarz connector Ds,tf(x,y) is identicallyO.

Proof. Since 8(0,0) = L:m,nEZam,n converges squarelikely, from proposition 1 it follows that the series for f(x,y) converges abso- lutely. Hence, for nonzero reals h and k the series for the Schwarz difference quotient Q(i\)f(x; y) converges absolutely. But, as in (5), Q(it,k/(X, y) is given by

Q(i\/(x, y) =

L

m,nEZ

By the absolute convergence, we can rewrite Q(si\)f(x, y) = lim p;/(x, y),

, N-+oo

Then, by the summation by parts formula proved in Proposition 2, we have

p;/(x, y) =

2: 2:

O$p$N(m,n)EB!,

+A(_N,_N)(X,Y)>'(~N,_N)+(h,k), where

A(m,n)(X, y) =

2:

(p,q)~(m,n)

and

6.>' (8,t_m,n)(h, k) =>.(",t_m,n)(h, k) _ >.(",t_m,n)+(h, k).

Note that each A(m,n)(x, y) is a squarelike partial sum ofSex, y), and so

by the hypothesis.

we thus have

lim A(_N,_N)(X,y) =0 N-oo

Since >.(",tm,n)(h,k) are bounded for small h and k,

(13) Q(h\)!(X, y) =

2: 2:

N,?-O(m,n)EBN

Now fix (s,t) E {O,l}2 and (x,y) E 1R^{2 ,} and let ^f > 0 be given arbi- trarily. By the hypothesis about the squarelike partial sums, we can chooseM >0 such that

(14)

whenever (m,n) EBN and N;::: M. Since limh,k_o6.>'(~,n)(h,k)= 0, lim "" "" A(mn)(X,y)6.>.(8~n)(h,k)=O.

h,k-O L..J L . . J ' ,

O$N$M(m,n)EBN

Hence we can choose a positive number 'T/ with 'T/ <l/M such that (15)

I L L

O$.N$.M(m,n)EBN

for all hand k with Ihl :::; 'T/ and Ikl :::; 'T/. On the other hand, by (14) we have

ILL

N>M(m,n)EBN

:::; L L

N>M(m,n)EBN

:::; f

L

L

N>M (m,n)EBN

:::; f

L

L

N~l (m,n)EBN

The last inequality follows from lemma 5. This result together with (15) implies

IQ(ht,k)/(x, y)1 :::;Cf

provided 0< Ihl < 'T/, 0< IkI < 'T/, and 1/2 :::; Ih/kl :::; ^2. Since ^f was arbitrary, we therefore concludeDs,tI(x, y) ==limh,k-+oQ(ht,k/(x, y)=

o. 0

§3. Proof ofthe main theorem

We first quote the following result from [AFR]. For the original version of the following proposition and the detailed proof the readers·

are referred to [AFR].

PROPOSITION 7 [AFR]. Let I(x,y) be a continuous function. Ifall theSchwarz connectors ofI(x,y)areidentically 0, then all the Schwarz differences of f(x,y) areidentically O.

Suppose thatSex, y) = L':m,nEZam,neimXeinyconverges squarelikely to 0 everywhere and the squarelike partial sums A(p,q) = L':(m,n)::::;(p,q)

am,neim:teiny, (p,q) E BN, are equal to o(N-¹⁾ as N -. 00. Then, proposition ¹implies that the second formal integral f(x,y) of S(x,y) is continuous, and proposition 6 implies that all Schwarz connectors of f(x, y) are identically O. From proposition 7 it now follows that

!:l~ht,k/(X,y)= ⁰ for all (s,t) E {0,1}2 and (h,k) E 1R^{2 -} {(O,O)}. In particular, !:l~~~,211')f(x,^y) =^O. But, !:l~~~,211')f(x,^y) =^ao,O' Hence we obtain the following result.

COROLLARY 8. Suppose that a 2-dimensional trigonometric series

S(x,y) =

L

converges squarelikely to 0 everywhere and allof its squarelike partial sums SE(X,y) are equal to o(/EI-¹⁾ as IEI -. 00. Then ao,O =O.

For each (p, q)E1.² let us define a transformation^T(p,q) which trans- lates a 2-dimensional trigonometric series :

( ~ ^{imx in y)} ~ ^{im:t iny}

T(p,q) L....J am,n e e = L....J am_p,n_qe e .

m,nEZ m,nEZ

PROPOSITION 9. If S(x,y) = L:m,nEZ am,neim:teiny satisfies the hy- pothesis of the main theorem, then T(p,q)S(x,y) also satisfies the same hypothesis for every(p, q) E1.².

Proof. Once we prove the assertion for the cases of(p, q) =(±1, 0), (0, ±l), then by the repeat of the same argument we can obtain the desired result. The treatments of the cases of (p,q) = (±1,0),(0,±1) are similar to each other. So we shall consider only the case of(p, q) =

(1,0).

Put

T(x,y) = T(l,O)S(X,y) =

L

where bm,n =^am-l,n, and for eachE C 1.² write SE(X,y) =^L:(m,n)EE

a_m,n^eimxeiny and TE(X_,y) = '"L..-(m,n)EE^bm,n^eimxeiny. Let ^€ > 0 be given arbitrarily, and choose M >0 with M-I ~ M/2 such that (16) ISE(X,y)1 <~ ^{if N} ~ ^{M and LN-l C E C LN.}

Suppose N ;::: M and £N-l C E C £N, and let F = ((m,n) E Z²1(m+1,n) E E}. Then

TE(X,y) =

L

L

(m,n)EE (m,n)EE

=eix

L

(m,n)EF Clearly £N-2 C F C £N+l'

Now put A = Fn£N_l,B= £N-l U(FnBN), and C= £NU(Fn BN+t}. Then

(17)

TE(X,y) = eix (SA(X, y)+SB(X, y)+Sc(x, y)-SCN_l (x, y)-SCN(X, y)).

But by (16),

ISA(X, y)1 < f/(N -1) :s; 2f1N,

ISB(X,y)1 < fiN, ISc(x, y)1 < f/(N+¹⁾< fiN,

and

ISCN_l(x,y)1 < f/(N-1) :s;2f1N, ISCN(X,y)j < fiN.

It now follows from (17) that ITE(X,y)1 < 7f1N. Since ^f > 0 was arbitrary, we now conclude that the squarelike partial sumTE(x,y) is equal too(lEI-¹⁾ as IEI-t^00. This also implies that T(x,y)converges squarelikely to 0 everywhere. We have thus proved the assertion for the case of translation7(1,0). 0

We can now complete the proof of the main theorem.

Proof of the main theorem. Let Sex, y) = L:m,nEZam,neimxeiny be a 2-dimensional trigonometric series satisfying the hypothesis of the main theorem. Then, for each (p,q) E Z2, the translate 7(p,q)S(x,y), whose constant term is ap,q, also satisfies the same hypothesis. Hence, by Corollary 8, ap,q = o. ^{Therefore, all} the coefficients am,n are equal to O. 0

§4. Epilogue

The main result is stated under the restriction (18)

where E is a squareli1ce indexing set. We can compare (18) with the following condition about square partial sums :

(19)

L L

Iml<r Inl<r

These two conditions are not equivalent. In fact, it is an open problem whether (19) can replace (18) in order to derive the main result. But, one can easily derive the following corollary from the main result.

COROLLAY 10. Suppose that a 2-dimensional trigonometric series S(x,Y) =

L

m,nEZ

satisfies

L L

Iml<rlnl<r

and

L

max{lml,lnl}=r Then all the coefficientsam,n are O.

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Department of Mathematics Ajou University

Suwon 442-749, Korea