Abstract. We survey some recent results on the uniqueness questions on multiple trigonometric series. Two basic questions, one about series which converges ...
Contemporary Mathematics
Uniqueness Questions for Multiple Trigonometric Series J. Marshall Ash and Gang Wang Abstract. We survey some recent results on the uniqueness questions on multiple trigonometric series. Two basic questions, one about series which converges to zero and the other about the series which converge to an integrable function, are asked for four modes of convergence: unrestricted rectangular convergence, spherical convergence, square convergence, and restricted rectangular convergence. We will either get into the details or outline some of the proofs for the known uniqueness theorems. Some results on the sets of uniqueness are also given. Finally, we will mention some interesting open questions in this area. Some of them are even one-dimensional. We assume the reader has some basic knowledge of measure theory and Fourier analysis. Most of the topics and materials can be understood by upper level undergraduate students.
Contents 1. Introduction 2. Some Cantor-Lebesgue Type Theorems 3. A Uniqueness Theorem for Unrestrictedly Rectangular Convergence 4. A Uniqueness Theorem for Spherical Convergence 5. Sets of Uniqueness under Spherical Summation 6. Questions about Square and Restricted Rectangular Uniqueness 7. Orthogonal Trigonometric Polynomials References
1 6 10 14 20 24 31 35
1. Introduction In 1870, Cantor proved the following uniqueness theorem for one-dimensional trigonometric series. 1991 Mathematics Subject Classi cation. Primary 42B99; Secondary 42B08, 42A63. Key words and phrases. Cantor-Lebesgue theorem, restricted rectangular convergence, set of uniqueness, spherical convergence, square convergence, uniqueness, orthonormal sequence, unrestricted rectangular convergence. G. Wang's work is partially supported by a grant from Tsinghua University, Beijing, P.R. China, a grant from the Chinese National Natural Science Foundation, and a grant from the paid leave program of DePaul University. c 0000 (copyright holder)
1
2
J. MARSHALL ASH AND GANG WANG
1.1 (Cantor). Suppose the one dimensional trigonometric P Theorem Pn series cn einx converges to zero in the sense that at each x 2 T = [0; 2 ); sn = k= n ck eikx tends to zero as n tends to 1: Then all cn must be zero. In other words, there is only one way to represent a trigonometric series. If two trigonometric series agree everywhere, then all the coe cients of the two trigonometric series are the same. The rst step of the is to obtain a growth rate condition for the coe cients P proof cn: : Observe that if cn einx converges for all x, then the n term of the series, c n e inx +cn einx also tends to zero for each x: The following Cantor-Lebesgue Theorem asserts that this happens not because of the cancellations between c n e inx and cn einx : Theorem 1.2 (Cantor-Lebesgue). If c q 2 2 then jc n j + jcn j ! 0:
ne
inx
+ cn einx ! 0 for every x 2 T;
This theorem was proved by Cantor in 1870. In 1905, Lebesgue proved a stronger version. This is why the theorem is so named. What Lebesgue proved is the following Theorem 1.3 (Lebesgue). If for a positive measure set E inx e + cn einx ! 0 for every x 2 E; then n q 2 2 jc n j + jcn j ! 0: (1.1)
T; we have
c
The above theorem can also be strengthened. See Ash, Kaufman, and Rieders (1993) for details. Now we give the proof of Theorem 1.3. Proof: By Egorov's Theorem, we can nd a subset E0 of E such that jE0 j > 0 and on E0 ; c n e inx + cn einx is uniformly convergent to zero. Thus, by Steinhaus' Theorem, there is a nonempty open interval I = ( a; a) such that I E0 E0 = fx y : x; y 2 E0 g : Let dn (x) = c n e inx + cn einx : Then lim Dn = lim sup jdn (x)j = 0:
n!1
n!1 x2E0
Given " > 0; choose N1 such that Dn < " whenever n N1 : Since ( a; a) E0 E0 for some a > 0; there exist N N1 such that = (2n) 2 ( a; a) for all n N: Thus, for each n N; there exist xn ; yn 2 E0 such that n (xn yn ) = =2: Consequently, from c
ne
inxn
+ cn einxn = dn (xn )
c
ne
inyn
+ cn einyn = dn (yn )
we have einyn dn (xn ) einxn dn (yn ) einyn dn (xn ) einxn dn (yn ) = ; 2 e in(xn yn ) ein(xn yn ) e inyn dn (xn ) + e inxn dn (yn ) e inyn dn (xn ) + e inxn dn (yn ) : cn = = in(x y ) in(x y ) n n n n 2 e e This implies q p p 2 2 jc n j + jcn j 2Dn 2"; which proves the theorem. We now prove Theorem 1.1. c
n
=
MULTIPLE TRIGONOMETRIC SERIES
3
Proof: We rst form the Riemann function x2 X cn inx F (x) = c0 e : 2 n2 n6=0
By the Cantor-Lebesgue Theorem, F is uniformly convergent. Thus, F is continuous. Next, observe for all x 2 R X F (x + h) 2F (x) + F (x h) = c0 + I (nh) cn einx 2 h n6=0 0 1 X X @ = ck eikx A (I ((n + 1) h) I (nh)) n 1
where
I (x) = Note limx!0 I (x) = 1: Since X
n 1
jI ((n + 1) h)
8 < :
jkj n 2
sin( x 2)
if x 6= 0
( ) x 2
1
if Z
I (nh)j
1
h
" for all > 0 : > > jx xi j< : ; xi 2T d ;i=1;2
18
J. MARSHALL ASH AND GANG WANG
It is easy to see that Z" is a closed set. The discontinuity set of F (x) in T d is [ Z= Z" : ">0
We will show that Z = ;. P Assume Z 6= ;. By assumption, cm eihx;mi is uniformly bounded in r at jmj r
each point x of T d , thus
8 [ \< x2Z: :
n 1k 1
X
jmj2
cm eihx;mi
n
k
By Baire's category theorem, for some n 1, 8 X \< cm eihx;mi x2Z: : 2 k 1
jmj
9 = ;
n
k
= Z:
9 = ;
has a nonempty interior relative to Z. Namely, there exists an open ball B(p; Z such that (4.11)
max k
max x2B(p;
0 )\Z
X
cm eihx;mi
0 ); p
2
n < 1:
jmj2 k
Consequently, F (x) =
X1 X cm eihx;mi = k 2
k 1
jmj =k
X
k 1
1 k
1 k+1
X
jmj2
cm eihx;mi k
is convergent uniformly on B(p; 0 ) \ Z. Thus, when restricted to B(p; 0 ) \ Z, F (x) is continuous. In addition, the arguments given in (4.8) combined with (4.11) show that (4.12)
jF (x)
A F (x)j
C
2
uniformly on B(p; 0 ) \ Z. We need the following three theorems to show that this will lead to a contradiction. The rst theorem generalizes a result of Rado. Theorem 4.6. Let F be a bounded function de ned on ball B(p; r) and Z be the set of discontinuity of F in B(p; r). Suppose for each x 2 B(p; r), (4.13)
~ F (x) = 0:
If for all x 2 B(p; r) n Z, the harmonic measure (4.14)
!(B(p; r) n Z; @Z; x) = 0;
then F is harmonic on B(p; r) provided that F is continuous when restricted on Z. For an open set G, the harmonic measure of F @G related to G at x 2 G, !(G; F; x), is closely related to Brownian motion. Let fXt gt 0 be the standard Brownian motion in Rd and ( ; F; P x ) the be corresponding probability space such that P x (X0 = x) = 1. Denote T be the stopping time of Xt hitting @G: T = infft 0 : Xt 2 @Gg. We use the convention that inf ; = 0. Then the
MULTIPLE TRIGONOMETRIC SERIES
19
harmonic measure on @G at x 2 G is the distribution of XT under P x . That is, for F @G, !(G; F; x) = P x (XT 2 F ) = E x IF (XT ); where E x is the expectation operator under P x . If f (x) is a bounded measurable function de ned on @G, then Z Z (4.15) Hf (x) = E x f (XT ) = f (z) P x (XT 2 dz) = f (z) !(G; dx; z) @G
@G
is a harmonic function in G. Moreover, if f is continuous on @G and @G satis es the exterior cone condition, for example, then Hf is continuous in G and converges to f at every point of @G. See Helms, for example, for details. An important property regarding harmonic measure is summarized in the following lemma. It follows almost immediately from the continuity of Brownian motion paths. Lemma 4.7. Let G be an open set and let the closed set F be containedTin G. Suppose Gk is a sequence of nested open sets containing F such that F = Gk , k 1
then for x 2 G n G1 ,
!(G n Gk ; @(G \ Gk ); x) # !(G n F; @F; x)
as k ! 1:
The second theorem, which is Bourgain's key contribution, is the following inequality. Theorem 4.8. Let F; B(p; 0 ) and Z be the objects described in the discussion preceding Theorem 4.6. If p1 2 Z; B(p1 ; 1 ) B(p; 21 0 ) and p2 2 B(p1 ; 21 1 ), then, (4.16)
jF (p1 ) F (p2 )j h c jF (p1 )j + +
3 4 (d
sup q2B(p1 ;2
1)
1
i
[1
jF (p1 )
1 )\Z
!(B(p1 ; !
1)
n Z; @(Z \ B(p1 ;
1
1 )); p2 )] 4
F (q)j :
The last theorem is concerned with general harmonic measure again. Theorem 4.9. Let B(p0 ; r) be a ball in Rd and F a closed set such that B(p0 ; r) \ F 6= ;. Suppose for some x 2 B(p0 ; r) n F , !(B(p0 ; r) n F ; @(B(p0 ; r) \ F ); x) > 0: Then there exists p1 2 B(p0 ; r) \ F , such that inf lim inf 1 >0
2 !0
inf
x2B(p1 ;
2)
!(B(p1 ;
1)
n F; @(B(p1 ;
1)
\ F ); x) = 1:
We will not prove these three theorems. Interested readers can read the survey paper written by Ash and Wang for details. Now we nish the proof of Theorem 4.5. If for all " > 0, !(B(p; for all x 2 B(p; !(B(p;
0)
0)
0)
n Z" ; @(B(p;
0)
\ Z" ); x) = 0
n Z" , then
n Z; @(B(p;
0)
\ Z" ); x)
!(B(p;
0)
n Z" ; @(B(p;
0)
\ Z" ); x) = 0;
20
J. MARSHALL ASH AND GANG WANG
for all x 2 B(p;
0)
n Z by the maximum principle. Thus, !(B(p;
0)
n Z; @(B(p;
0)
\ Z" ); x) = 0 S for all x 2 B(p; 0 ) n Z and all " > 0. Since Z = Z" , we have ">0
!(B(p;
0)
1 2 0
n Z; @(B(p;
0)
n Z" ; @(B(p;
0)
\ Z); x) = 0:
Let p1 = p and 1 = in (4.16) to see that F is bounded on B(p; 40 ). Since it is also continuous when restricted to B(p; 0 ) \ Z; (4.9) and the above imply that the hypotheses of Theorem 4.6 hold. Thus, F is harmonic and hence continuous on B(p; 40 ), which is a contradiction. Therefore, for some " > 0, we must have !(B(p; for some x 2 B(p; that (4.17)
0)
\ Z" ); x) > 0
n Z" . By Theorem 4.9, there exists p1 2 Z" \ B(p;
inf lim inf 1 >0
0)
2 !0
inf
x2B(p1 ;
2)
!(B(p1 ;
1)
n Z" ; @(B(p1 ;
1)
0
8
) such
\ Z" ); x) = 1:
) is continuous, there exists 0 < 1 such that " (4.18) jF (y) F (p1 )j for all y 2 B(p1 ; 2 1 ) \ Z: 10 Let > 0 be any positive number. By (4.17), there exists 0 < 2 = 2 ( ) such that Because F restricted to Z \ B(p;
(4.19)
!(B(p1 ;
1)
0
8
n Z; @(B(p1 ;
1)
\ Z); y) > 1
for all y 2 B(p1 ;
2 ):
Since p1 2 Z" , by de nition, there exists p2 2 B(p1 ; 2 ), such that " : jF (p1 ) F (p2 )j 2 Applying Theorem 4.8 with 1 = 1 , by (4.18)-(4.19) and the above inequality, we have 3 " " (d 1) 1=4 jF (p1 ) F (p2 )j [jF (p1 )j + 1 4 ] + : 2 10 This is also seen to be a contradiction by letting ! 0. Thus we completes the proof given the above three theorems are true. 5. Sets of Uniqueness under Spherical Summation This section exposites work we have done in two papers.[AW5, AW6] One motivation for studying spherical uniqueness under Abel summability is to study the set of uniqueness question. Definition 5.1 (Set of Uniqueness). A set E T d is called a set of uniqueness, or U -set, if every multiple trigonometric series P spherically convergent to 0 outside E vanishes identically. That is, if S (x) = cn eihn;xi converges spherically to zero for all x 2 T d n E; implies that cn = 0 for all n 0; we say set E is a set of uniqueness, or U set. Thus, by Theorem 4.2, we know the empty set ; is an U set. Interesting questions are: Can an U set be nonempty? Can it be countable? Can it be uncountable? Can it have positive measure? And more can be asked. To study sets of uniqueness is to investigate if one can weaken the hypothesis of Theorem 4.2 while achieving the same conclusion.
MULTIPLE TRIGONOMETRIC SERIES
21
Historically, the major tool for studying sets of uniqueness is formal multiplication. P 1 ihn;xi 1 Definition 5.2 (Formal Multiplication). Let S (x) = cn e and S 2 (x) = P 2 ihn;xi cn e be two multiple trigonometric series. The formal multiplication of S 1 P 2 ihn;xi and S is another multiple trigonometric series S (x) = cn e ; where X 1 2 cn = cn v cv if such sum converges for each n and the summation method is the same method used in the original series.
In general, we formally multiply a convergent series by the Fourier series of a smooth function. Thus, the growth rate of the second series coe cients c2n are rapidly decreasing to ensure the converges of the coe cients cn given in the de nition. When d = 1, formal multiplication by a smooth function preserves convergence. In higher dimensions, we only know that formal multiplication by a smooth function transforms convergence to Bochner-Riesz summability of a certain order. P Definition 5.3 (Bochner-Riesz Summability). We say a multiple series n an is Bochner-Riesz summable to A of order 0, denoted by summable (B R; ) to A; if X lim an (1 jnj2 =R2 ) = A: r!1
jnj r
P P Remark 5.1. It is easy to see that if n an is summable P (B R; ) to A, then 0, and n an is also spherically n an is summable (B R; + ") to A for any " Abel summable to A. Since Abel summability has its natural connection to formal multiplication, it is useful to give a de nition for set of Abel uniqueness. To give a reasonable de nition of set of Abel uniqueness, we need to pose certain conditions on the growth rate of the trigonometric series. Recall when d = 1, the example of 0 (x) = P n sin nx shows that condition cn = o(jnj) is necessary for the empty set to be a set of Abel uniqueness. But, a more restrictive condition than cn = o(jnj) is needed to avoid P the empty set being the only set of Abel uniqueness. Consider (x) = 1=2 + n>0 cos nx. Note that is Abel summable to 0 except at the origin. Thus, for d = 1, we need a growth rate condition like (5.1)
cn = o(1)
as jnj ! 1
in order to include singletons as sets of Abel uniqueness. We therefore give the following de nition. Definition 5.4 (Set of Abel Uniqueness). A set E T d isPcalled a set of Abel uniqueness, or UA -set, if every multiple trigonometric series cn eihn;xi with coe cients satisfying X (5.2) jcn j2 = o(r2 ) as r ! 1; and cn = o(1) as jnj ! 1; r=2 jnj 0. Since every positiveSmeasure set has a closed subset of positive measure, we may assume that E = [E + 2 ] is closed in d . Note that the indicator function of E, E , is in L2 (T d ). Thus, the coe cients cn of the Fourier series of E satisfy condition (5.2) since X (5.3) jcn j2 = jj E jj2 = jEj < 1: n
Because E is closed, the Fourier series of E is spherically Abel summable to 0 o E by a Theorem of Shapiro. Thus, cn = 0 for all n since E is a UA -set. Consequently, jEj = 0 by (5.3), which is a contradiction. Early works of Shapiro are the only strong multi-dimensional results on sets of uniqueness. Using Theorem 4.4, we can generalize these results to higher dimensions. We rst show any singleton is a set of Abel uniqueness.
Theorem 5.2. Let q be a point in T d and f be a integrable function which is everywhere nite except possible at q: If a multiple (d 2) trigonometric series P cn eihx;ni is spherically Abel summable everywhere to f except possibly at q; and P if the coe cients fcn g satisfy condition (5.2), then cn eihx;ni is the Fourier series of f (x). P inx Note the theorem is false when d = 1 since the trigonometric series e is Abel convergent to 0 everywhere in T n f0g. Since spherically convergent implies spherically Abel convergent, we have the following corollary. Corollary 5.3. Let q be a point on T d and f be a integrable function which is everywhere nite except possibly at q: If a multiple (d 2) trigonometric seP ihx;ni ries c e is spherically convergent everywhere to f except possibly at q; then n P cn eihx;ni is the Fourier series of f (x).
Therefore, any singleton is also a set of uniqueness. We now extend sets of uniqueness from singletons to countable and uncountable sets. Using Theorem 4.4 and formal multiplication of trigonometric series, it can be shown that Theorem 5.4. For any d, H J sets are UA -sets. Theorem 5.5. Suppose that fEk gkS1 is a sequence of UA -sets, and for each k, Ek is closed in the sense that Ek = [Ek + 2 ] is closed in T d . Then E = S k 1 Ek is a UA -set.
MULTIPLE TRIGONOMETRIC SERIES
23
The de nition of an H J set is quite complicated. Definition 5.5. For any J 1, we say that fnk gk 1 = (n1k ; ; nJk ) k 1 is a one dimensional normal sequence of degree of freedom J if for every h = (h1 ; ; hJ ) 2 Z J n f0g, there holds lim h1 n1k +
(5.4)
+ hJ nJk = lim jhh; nk ij = 1: k!1
k!1
Say that fn1k ;
; ndk gk
1
= f(n11k ;
; nJ1k ; n12k ;
; nJ2k ;
; n1dk ;
; nJdk )gk
1
is a d dimensional normal sequence of degree of freedom J if each of the d sequences fn1k gk 1 ; , and fndk gk 1 is a one dimensional normal sequence of degree of freedom J. Say a set E T d is a set of type H J if there exists a d dimensional normal sequence = f k gk 1 of degree of freedom J and a nonempty domain D T dJ such that k x 2 = D mod 2 for all k 1, where x=f =
k
xgk
f(n11k x1 ;
1
; nJdk x1 ; n12k x2 ;
= fx1 n1k ; x2 n2k ;
; xd ndk gk
; nJ2k x2 ;
; n1dk xd ;
; nJdk xd )gk
1
1:
For example, the Cantor set is an H 1 set in d = 1 corresponding to = f3k g and D = (1=3; 2=3). The d-th power of the Cantor set is also an H 1 set. Moreover H J sets are strictly increasing sets. That is, there is a set E which belongs to H J+1 ; but is not in any of the lower H K sets for K J. It can be shown that in general the complement of an H J set is dense in T d , hence has full measure and is of the second category in the sense of Baire. Therefore, by Remark 2.1, we get the following corollary. Corollary 5.6. For any d, H J sets are U -sets. Theorem 5.4 and Corollary 5.6 answer a question raised by Shapiro more than twenty years ago. Combining Theorem 5.2, Theorem 5.5 and Remark 2.1, we have the following corollary for set of uniqueness and set of Abel uniqueness: Corollary 5.7. Any countable set is a UA -set and a U -set. Since the product of the one dimensional Cantor set with the d 1 dimensional torus is an H 1 set in dimension d, by Theorem 5.4 and Corollary 5.6, there exist uncountable UA -sets and uncountable U -sets. We now outline brie y how Theorem 5.4 is proved. Let H be a H J set. Its geometric structure allows us to construct a sequence of C 1 (T d ) functions f k (x)gk 1 , of period 2 in each variable, having the following properties: S (1) For each k 1, there exists a set Dk = [Dk + 2 ] open in T d such S that H = [H + 2 ] Dk and k vanishes on Dk ; (2) For each k, the Fourier coe cients of k , f nk gn , satis es nk = o(jnj (8d+4) ) as jnj ! 1; P (3) There exists a nite constant C such that n j nk j C for all k 1 (4) limk!1 nk = 0 for n 6= 0; (5) limk!1 0k = 1
24
J. MARSHALL ASH AND GANG WANG
This was proved by Shapiro. Now suppose that X (5.5) lim+ cn eihn;xi jnjt = 0 t!0
d
for all x 2 TPn H and that fcn g satis es condition (5.2). Form the formal multiplication of cn eihn;xi with the Fourier series of k (x) : If we let f nk g to be the Fourier coe cients of k (x), then the formal multiplicationPhas coe cients Akn = P k c n . Since k (x) vanishes on Dk H and limt!0+ cn eihn;xi jnjt = 0 for all x 2 T d n H , it is plausible to have X (5.6) lim+ Akn eihn;xi jnjt = 0 for all k 1 and all x 2 T d : t!0
n
In fact, this can be shown rigorously. Next, since the formal multiplication by smooth functions like k preserve the condition (5.2), for each k 1; coe cients Akn also satisfy Bourgain's condition. Now apply Theorem 4.4, we have Akn = 0
(5.7) for all k (5.8)
1 and all n 2 T d . We show this leads to cn = 0
d
for all n 2 T . Fix n. Condition (5.2) implies that cm = o(1) as jmj ! 1. So, for any " > 0, there exists n0 > 1 such that jcm j < "=C for all jmj n0 =2 and jm nj jnj=2 whenever jmj n0 , where C is the constant given in property (3) of the above list. Then, condition (5.7) and property (4) above imply that X k j 0k cn j " + jcn m jj m j: 1 jmj n0
Let k ! 1. By properties (4) and (5), we have jcn j ". This proves the theorem. In the next section, we will discuss what we know about the uniqueness question for square and restrictedly rectangular convergence and then pose some open questions. 6. Questions about Square and Restricted Rectangular Uniqueness Uniqueness questions for multiple trigonometric series under square and restricted rectangular convergence have been of interest for only some 30 years. It could have been longer. In 1918, a general uniqueness result on rectangular convergence was announced. It seemed to have a routine inductive proof, so interest in the subject died out. About 1970, Ash and Welland realized that the proof was incorrect. The rectangular problem then became interesting again. But, people also realized the problem was a hard one, as we have seen in the earlier sections. The problem for square or restricted rectangular convergence is still open even for dimension d = 2: A major reason why this is so hard is because, as shown in Section 2.2 and 2.1, Cantor-Lebesgue type theorems fail. Without this condition, the Riemann function will not converge in general, so we fail at the analogues of the very rst step of all the established proofs of uniqueness. One may try to construct counter examples to show the uniqueness theorem here may be in fact false. But this also seems to be very di cult. We will explain in the following the several leads that we have tried for this problem and some interesting questions arising from our investigation.
MULTIPLE TRIGONOMETRIC SERIES
25
We note rst that restrictedly rectangular convergence implies square convergence. So when trying to prove a uniqueness theorem it is better to assume restrictedly rectangular convergence; while when trying to construct a counterexample, it may very well be easier to nd a non trivial series that is square convergent to zero everywhere. At the very least, a contradiction to square uniqueness is a necessary step on the way to nding a counterexample for restricted rectangular convergence. The counter example we constructed in Section 2.1 for Cantor-Lebesgue type of theorem is not a counter example for square uniqueness. The reason is the limit function is not in L1 : If it were, then this would have disproved the L1 uniqueness theorem. To see why the limit function is not in L1 ; note that the series is a \butter y" shaped one, namely the non zero indices of the series lie on a \butter y" region. We know for any \butter y" shaped series, square convergence means iterated convergence, which would give uniqueness results by induction. This means Cantor-Lebesgue type of results would hold true. So any counter example for square convergence has to be a one which does not converges iteratively. But we cannot nd such one yet even to counter the Cantor-Lebesgue theorem. The example we constructed is really a one dimensional one. It is very hard to construct interesting purely multi-dimensional convergent multiple trigonometric series other than Fourier series. Lack of examples is what makes this eld so mysterious. 6.1. Three weak theorems. We'll begin with three weak square uniqueness theorems. The rst two theorems hold in a much more general setting and have quite easy proofs. None of the three theorems amounts to very much, but better results have yet to be found. Theorem 6.1. Suppose Sm =
X
cn e2
ihn;xi
n2Zd
converges to 0 uniformly on T d . Then
cn = 0 for every n 2 Zd . Proof. For each n 2 Zd , we have Z d cn = (2 ) Sm (x) e
2 ihn;xi
dx
Td
as soon as
m max fjni j : i = 1; 2; : : : ; dg . Thus, interchanging limit and integral, we see that for each n 2 Zd , Z d cn = lim (2 ) Sm (x) e 2 ihn;xi dx m!1 Td Z d = (2 ) lim Sm (x) e 2 ihn;xi dx Td m!1
= 0:
Theorem 6.2. Suppose S=
X
n2Zd
cn e2
ihn;xi
26
J. MARSHALL ASH AND GANG WANG
is square convergent almost everywhere, for almost every x 2 Zd , Sm (x) ! 0 as m ! 1, and furthermore that
X Zd
Then
2
jcn j < 1.
cn = 0 for every n 2 Zd . Proof. Say E is the subset of Td of full measure so that whenever x 2 E, Sm (x) ! 0 as m ! 1. By the Riesz-Fischer Theorem, there is a function f 2 L2 Td so that the Fourier coe cients of f , f^ (n) = (2 )
d
Z
f (x) e
2 ihn;xi
dx
Td
satisfy
f^ (n) = cn for all n 2 Zd .
As m ! 1, we have Sm [f ]
f
L2 (Td )
X
=
f^ (n)
2
fn2Zd :some jni j>mg
X
=
fn2Zd :some jni j>mg
2
jcn j ! 0,
where Sm [f ] denotes the square partial sum of the multiple Fourier series of f . This implies that there is a subsequence of positive integers m1 < m2
