On a Weyl Multiplier for Multiple Fourier Series
On a Weyl Multiplier for Multiple Fourier Series
Jau-D Chen
Department of Mathematics College of Sciences
[Abstract]
The purpose of this artíc1e is to explore a Weyl multiplier for the multiple Fourier series of a certain c1ass of complete orthnormal systems. It contains our previous results for a sufficient condition of almost everywhere convergence of the rectangular partial sums for both cases of multiple trignometric-Fourier series and multiple bounded generalized Walsh-Fourier series.
1. Introduction
ln 1930 Kaczmarz [4] showed that the condition C用
~
I
cQkI
210g (Q+2) log (k+2) <∞ 兒， k::::O . 自is sufficient for almost everywhere convergence of the rectangular partial sums of the corresponding double trignometric Fourier series. The generalization of Kaczmarz's theorem for both cases of multiple trignometric Fourier series [2] and multiple bounded generalized Walsh F ourier series [3] were given by Chen and Shieh. As a language of Weyl multiplier defined by E. M. Nikisin [幻， it implies that log(m+2) is a Weyl multiplier for a1most everywhere convergence, respectively, of multiple trigonometric Fourier series and multiple bounded generalized Walsh Fourier, where m :::: (m}, m2,.. • , mn) is an integer lattice point with mj ~ 0 for each i :::: 1, 2, . . . , n, and Log(m+2) :::: log(m} +2).
log(m2 +2). . . log(mn +2). For given n complete orthonorma1 systems of functions l ￠ fj}}∞( 'l'mk ] m k=O in L 2 (Ij) with period (bj-句)， where Ij:::: [aj , bj ] for each j
=
1,
2. . . . ，几 thenwe obtain a complete orthonorma1 system of functionsBulletin o[抽tionaZTaiwan NormaZ University No. 28
I (1), , (2) _ , (n) , , I
吵 = tþ~~' (Xl) 川 (X2) .... <Þ~: (Xn)
J
inU
(Tn) [肘， where Tn =圳 2X
. . x1n' Therefore, we can consider the n-tuple 1þ干ourier series of functions f(x) = f(Xj. 女ρon R n of period (bf
可)in each Xj, j = 1,2, . . . ,n. Now, it is natural to ask what a Weyl multiplier for such multiple 1þ-Fourier series is. ln this article, we will answer the question by showing that1
Lo伽叫} is also a Weyl multiplier for a suitable class of 1þ-Fourier series.2. Definitions and Notation
( . (j) 1 ∞ 9
Given the n systems of functions
f
<Þ dì~'\
~， _ A in U (Ij} with periodl r "'K J mk = 0 - '-J
(braj), where 于[旬， bj], j = 1, 2, . . . , n, w:e may consider the n-tours Tn = 11 x12x
...
xIn = [at, bt1
x[a2, b 2 ]x. .'. x[an, bn
1
ofpoints X = (x t, x 2,.. Xn). For f e L2 (Tn) let SQ(X 、 0 ， σ 史(X ，0
be respectively the rectangular pa{1:ial sum and the 缸> 1 )-sum for the n.tuple 1þ-Fourier series of f at x, where Q = 叭，缸， • . • , Qn) is a nonegative integer lattice point of R n. Analogous to usua1 multiple trigonometric Fourier series,are the following standardequa1ities
yvd AUAU 、‘， J 、‘， J VdV4 xx itit nvAnνA DK 、‘.'，、‘ .J YY W 叭 W 叭 nn TT rlJrlEJ == ηn xxj ra ﹒、， s •• 、 nvAnνA QUG where 、y 、hj nn vdVd nn xx ' .. 、、'，區、 、.，'，、‘， J
“
!nhwEn 叫 K 、 EY 、‘ EJ '.句 a Ly 、“句'， xjXi r--、，.、 、‘，'，、‘，'， 句自 2 弓，占 2 , .. 、 nVAt-‘ nUA DK 、.. EJ 、‘ EF -A ••• vdVe --xx rstr.-、 、‘， J 、‘.'， ••• Aeah4.EA--itnEra 、 nUA DK == 、.. J 、‘ .J YVd XX JS ‘、 rs ﹒、 nEnE DK andUL
~j
.UL
,-;- (j) DL(呵 'Yj) = ~~ <Þk '-(J (Xj)(Þk ~J (yj)' k 司。。L
, (
~j ~
U) /
" /
KEj(X的 )=.itfk 1月 'Yj)}1 (伊 1)， for j = 1, 2, . . . , n It is convenient to have .the following definition which has been used'曲
[ 1, 2, 3] , repeated here fore corlvenience. Definition 1. Suppose that /1 盯I叭
.叮， k吋} into {1, 2, . . ., n} and /1' is a strict1y 卸creasingfundion from {1, 2, . .:.,
n吋 into
{1,.2,. .,n}such 伽t(V1 ， v2 ，.， vk ， hvx ，...汁'n.k
i
= {1, 2,On a Weyl Multiplier for Multiple Fourier Series It is convenient to have the following notation. For points X = (Xl' X宜，... ,
Xn) in Rn, for each i = 1, 2, • • • , n, we designate points in (n-l)-sapce with coordinates (X 1 ,.. • , Xi-l, Xi+ 1 , . . . , X
n
) 的焉， and for i<
j, points in (n-2)-space with coordinates 的，.. .， X仙 Xi
+1
'. . .，呵-1'Xj +1
" ，叮 X~ as Xij . Thus a point X in Rn may be desginated as (吭，其) or (Xi
' 呵，支ij) for i, j = 1, . • ., n. Wealso write dx = dx ， d支. = dX.dx.d五;;'-1 - --J ---IJ For any interva1 J C Rn we may write J = J;x .
J; = J;xJ;xJ;;J lJ ' , where J; is an (n-l) interv-- - - - "1 a1 and J;; an (n-2) interval. We adopt the obvious extensions of the notion, we can designate the points X in Rn as (xp '
X,,'), , _.- --dx as dX"dx"-"P-"P' ---- ---- ---. -, and the interva1 J in - --- - -Rn as - - " J.~xJ.~;kfor p ..- p each complementary pair 紗 ， p').
With the above notions, for points x, y e R n and for nQnnegative integer lattice points Q = (見 1, Q2, . . , Qn)' we designate
、.. , J μJ Y' "恥 J X rst 、‘'， 1.JHFJ nνA 吋 rk ••••• a LK 唱“ n= ••• J = HV 、.. J Vd x rs ﹒、 v nz k n.k (V'J
DEEJ(XJL=piD~fl勻， ypj)
ln the following, we recall the meaning of a Weyl multiplier [5) for the
com-plete orthonormal
system 收刊 dkx1) 叫小X2)'
..心1;
(Xn) } of functionsin L2 (Tn).
Definition 2. Suppose that ~ 入史=λ( 史 1. . . . ，肉，.. .，如 ~ is a .sequence of
posìtive numbers nondecr叫均已烏 for each fixed 有 Then '{ À Q } is called
a Weyl multiplier for convergence almost everywhere of series in the system
.(1). , (2) _ . (n). , 1
ψ=
1
cþζ(x. ml ' ... m2 1 )CÞ '.:.~ (X2) . . ，--~， .φ(x~) \ if the convergence of the series. • - r mn'---n' J
2 I
~I
Q
an Iλ Q < ∞IQ I
ìmplies that the limit
C1) /_. , ,(2) A. ~n)
2aEh1(X1)φ史2φEn(xn)
exists almost everywhere on Tn, whe時 the convergence is in the sense of Pringsheìm, that 函，summed by rectangles.
Also, we need the following notions.
Bulletii1 o[ National Taiwan Normal University No. 28 、‘ aF a--A fs ﹒‘、 If f e L 2 (Tt) and
v is defir叫 a~:" Definition 1, set
l
SQv (x v' f) 1j
Log(且+2)S已 (X
v
' η= supE P
. (2)
。九 (XJ=s;;l
uf(XVAland for g e L 2 (Tn), (ν ， V') is a complementary pair defined in Definition 1, set
= sup Qv KV T 且 riIJ KQ.. 伏， Y) ，ßYII sup Q""
{1fTn;kDEV 川川 j|
(3) p* 仰， (s， g)where TK=Ivx...x IV ,TnJK=Iu, x..X IU'
~1 ~k V VJ V
n_k
In what follows, A wi1l denote an absolute constant, which may vary from line to line.
3. Main theorem
In view of the proofs of our previous works [2, 3], the following operator inequalities (4) 一 (6) ，especially (4) and (5), play an important role:
(4)
11 門(X
v
' η112 ~A
Ilf11 2
, fe t.Z(T~)，
、‘'， kv 可 河 T a--可 rz 、 呵， ILZ L C •• aiv nbE 且 '。 b m' ,, feal--』 2 "+EA qb AA << 司，世弓，-η 的 vx xr( IIV MUU bhv σ DE 、 -E/ 、.. ', ZJfo /a ﹒、'，.‘、So, it is reasonable to consider the class of complete orthononnal system
吵 =(4JK1) 叫你2)
叫“n)
} offu叫少 inuqn川or
whichthe operator ineQualities (4) and (5) hold
,
whether ,1
Log(m+2) ~ is its Weylmultiplier or not. We give an affirmative answer First
,
we'need the following lemma.On a Weyl Multiplier for Multiple Fourier Series
(扒什吵川
VvJ= 〈
r1ρ)川(x )￠JJf(
伊的州
v內蚓2ρ~)
v
叭1 \~V1 ~'I' m
V
"'V2' • • .νk v'Vk ' Jany g defined as 旭 Definition 1. Then for aoy f e U (Tn) and any complemen-tary pair (v. v'), we have
I\P勻， (見。112 ~Allfl12
Proof. Except the notational complexities, under our assumptions the proof is essentially the same as Lemma 2 in our previous work 口， p.393].
Now we can state and prove our main theorem.
( . (1) , , .(2) f • (n) L _ ~ 1
Theorem. Lte 吵 = l 九; (x
1
)中m2(X2).. ￠IIEK(九)J
be a complete orthonormal system of functions in"L2 (Tn) such that the operator inequalities (4) and (5) hold 伽 an叮y systerri. 吵仇v
叫i
誌s a Wey抖1 mu叫I址ltip1i叮er for n卜叫"吋tuple 吵-Fourier series of functions 詛 L2(Tnη)
，
summed by recetangles.^
Proof.It sdfices to prove if f E LZ(Tn)and fm is the m-th Fourier
coefficient of f, suppose that
~ ICI 2Log(m+2)< ∞，
叮1 日..
then the rectangular
part
凶sums
{ S Q (x, f)J
conve
耶
almost
everywhere Let 們*吋S*f(x)=
sup I S史
Q(x咒，
ηI. By noting that t1叭ys
叩o applying the n-dimensiona凶a祉1 Abel partial summation formula i泊n [1] and the
t techniques used扭[口2訂]，we can ∞conc1ude t血ha前
們
11
s*f11 2~
A(;"f~IILog
川)
Finally, note the density of the 吵-polynomia1s in L 2 (Tn). So from the inequality (吋itfollows that
Illim sup
ISQ
紋，
η苟且，
(x, f) Ib
=o.
見，兒，
This implies the convergence almost everywhere
on Tn
,
so the proof is complete. ‘549-Bulletin of National Taiwan N01咿呀1 Universi伊 NO.28
In our previous work [2, 3] we showed that the conditions (4) and (5) in the theorem hold, for bo!h cases of the mutiple trigonometric-Fourier series and the mu1tiple bounded generalized Wa1sh Fourier series. It is a1so easì1y seen from the combinations of the proofs for those two theorems, and the conditions (4) and (5) are sti11 true for the cases of the multiple bounded genera1ized Wa1 sh-trigonometric Fourier series. Therefore our theorem remains more extensive for mu1tip1e Fourier series.
Fina11y, judging from a theorem of a Weyl multiplier for double trigonometric system [幻， we know that the Wey1, mul,tiplier in our theorem
cannot be essential1y improved.
REFERENCES
1. Chen, J au-D.
“
A Theorem of Cesari on Multíp1e Fourier Series."StudíaMath.
,
49 (1973),
69-80.2. Chen, Jau-D. and Narn-R. Shíeh
“
On a Sufficient Condítion for the Convergence of Multiple Fouríer Seríes." Bull. of Acad. Sinica. 5 (1977), 391-395.3. Chen, J au-D. "A Theorem of Kaczmarz on M u1tiple Genera1ized Wa1sh Fourier Series." Bull. of Acad. Sinica. 10 (1982), 205-212.
4. Kaczmarz, S.
“
Zur Theorie der Fourierschen Doppelreihen." StudiaMath. 2 (1930)
,
91-96.5. Nikisin, E. M.
“
Weyl multipliers for Multiple Fourier Series." Math.USSR Sbornik. 18 (1972), 351-360.
6. Zygmund, A. Trigonometric Series. vo1. 1 London: Cambridge Univ. Press, (1959), p. 34.
