'Sci.Rep.K
anaz
品，w
aUniv.
，Vol.1
5No.2，51c‑59
D巴C色mber1970

0:0 Divergenceof Fourier SerI:es，M
ichitakllKOZIM.ADetart
制 ent01Mathematics，Faculty01S
cience，Kanazawa
Univers
ity，Kanazawa 920(Received2
9October1970う

1.Introduction
Y.M.Chen[2Jhasprovedthatthereexistsafunctionoft
hec
lassL(log+log+IJ)Hwitht
hea
lmos
teverywherediv
ergentFour
ie
rser
ie
s by constr
uct
io
ns
i
mi
lart
ot
he
we
l
lknownc
lass
ica
lexample.
Thepurposeofthi
spaperistoshow t
hatt
heree
xist
s an everywherediv
ergentlog+log+L)l一
三 byY.M. C
hen'sresult[1J withrespecttothebestFourie
rser
ieso
fL(possibilityoft
heorderofthep
artialsumo
fFour
ie
rser
ieso
fL(O，2π).
同2.Themaintheorem
Wes
hallg
iv
eth
efollow
ingt
heorem.
.(t)，φ(t)bedefine
dforb~O andsatisfythefollow
ingconditions;ぇ;
THEOREM.L
et1(i) ，l(t)ispositive，decreasingto0，convex，and ~tì.' (t) isslowlyvarying.
(ii) Thereexistsaf
unction g(的 εL(O，21'1:)suchtha
tforeverypoint XE[O，21:)，む(y)(x)>'}AA'¥ forinfinitelymanyn.

l，(n)

(iii)
φ(t)isnon~negative andt
φ(t)isincreasingandconvex.
(iv)

1，...， 1，̲，， 1ー(五)2，l'(五)φ((五)1日 )EL(O，2π)forsomeδ>0.

hereexistsa function 1(的 ELφ (L)(0，21'1:)such t
hattheFouriεrseries
Then，tdivergesforallXE[0，21'1:).

，3.Proofofthetneore
殿Weputrp(x)= L:.
l(j)cosjx‑ 57‑

58Mi帥Hak~J KOZIM/"vv
herethis企rgesforall必 巴2釘)toafunctionconv
ecausedecreasestり oandi
8convex。
Fourierser
ieso
f め b
Nowweconsiderthefunctionandi8the‑9」
ごπ2¥=(判的 (Z)=:JJn.
10"'"
eshaHShOVl thatthisfunctionf(xiis(1) TheFour
台 rserIesofdiャergesfora
1
1X E[0，thereshouldbeaXijE[0，convergesandl
et

ベ
"sS"

for i
弓と 01U"for
i=0，1or J=
ThenbyAbel'stransfor
mat
ion
D.
.

，，‑";:;Cn，jS
j十 0‑Sj‑I)十 O=8j~l

1一一一 τL}for lz
三h玉n‑1)1) 伊 (j〉
jミ;;00forj，;?nwhereSincethenlatrixsatisfiestheToeplitzcondition，→ 8‑8=む → ∞ ) .

Ontheother.
1 ~，~'
2 ー~Àω 一一一-=一一­

A>9.
.
.

叫fori
n
fin
itel
ymany払 80wehavet
hecontradiction，(2叫 E三I
nor
dert
oprov
ethis，asnlvt
oshowthat
wen出 doand conv
e玄，

18

ELφ

Since め d邑creas
邑sto0and‑tJ.
'(t)isslowlyvarying，thenasx‑十 O[3J，p.

Therefore259JUZ︑︑︑︐︐︐︑︑︐︐︐
zoe︑r

φ

∞mt‑i。( ‑ 7〉(7)EY(7〉@((ーす〉 (7)ZY(7〉 dz

孟

27tπr，.
"，
1const.¥‑‑(一一~ )(一1一)
2，i'
(一一)
O
(const. (ーァ)
1吋 )
dxrt

孟J0・Z︐︑︐︐︑‑aauπjOnDivergence01FourierSeries2ππrJo':.:"

1，.
.
.
， 1 π 1，.
.
.
， 1x''x'.x

π21，r.
，.
， 1〉，̲" 1豆 const.¥‑‑(ー 1〉(‑z)ZY(‑zφ((‑zps)dzく ∞ .

Q.
E.
D.

4.CoroIlaries.
Givenanynumbere(0<e孟 1，)theree
xist
saf
unct
ionoftheCOROLLARY1log+log+L)ト εsucht
hatt
heFourierseriesdiv
ergesev
erywhere.
c1assL(uchthat0く可 <e.Weputi，(t)口(loglogt)ーはーが fort逗 8PROOF.
Wechoose可 sandextenditt
obeconvexfort
孟0，andputo(t)= (log+log+t)
l‑o.Thent
hecon
ditions(i) and(iii)ar
esatisfi巴d. Thecondition(ii)isY.M.Chen'sremarkableresu
lt[lJ.
Since1，.
.
.
， 1，̲" 1，.
.
.
.

1 ，，，，
̲̲ー(一二一)2，ì'(-:了)Ø((~) 1+ò) 話 const. 一一一 (log~ )l(loglog‑::‑〉
一品 晶
x'x "X ‑ー{1+(ε ーの}asx.
→ +0，thecon
dition(iv)i
ssatisfied.
. I
fforanyg
iv
enpositiv
esequence
μ，，= 0(logn) thereexistsa
COROLLARY2ucht
hatforver
ypointXE[O，21r)functiong(x)ELsforin
finitelymanyn，
thenforanygiv
ennumbere(Oく ε孟 1)，thereexistsafunc
tionofthec
1assL(log+L)ト εsucht
hattheFou
r
ie
rser
ie
sdiv
er
ge
sev
er
yw
here.
TheconditionofthiscorollaryisknownasZygmund'sconjecture.
PROOF.Asbefore，wechoose可 suchthat0<甲<e. We puti，(t)=(logt)一<1‑7，> fort
ミ2andexten
dittobeconvexfort~o ， andputO(t)=(log+t)l‑e. SinceSη(g)(x)>p.
"ー(一一)
2，i'(一一)O((一一)1+り孟 const.~ (log~)
X'XX

ー{l+(oーη)}

asX→ +0，thereforetheallconditionsofthetheoremaresatisfied.

REFERENCES(1) Y.M.CHEN，A remarkabledivergentFourierseries，Proc.Japan.A
cad.
，38(1962)，
23
9‑
244.
(2) Y.M.CHEN，Analm6stev
erywherediv
ergentFourierseriesofthec
lassL(
1og+1og+L)
1‑'，

Journ.London.Math.Soc.
，44(1969)，643‑654.
.ZYGMUND，Trigon仰 倒 的 series，Vol.1，Cambridge，1959.
(3) A