Sd. Rep， f!
UtmliZllWa UI
1IY，V0.
1a2， 1¥"
1o.2，pp.81~-- 86
Dec.1987

Typeunl.
'VHcnkinF'ouxic1On

Conver:g'ent

YushilmzuUNOoj}Cana
Z
{l'U
Ja，

(R.eceiv
edOctob宕r
19，1987)i¥.bst!.'品 d. W巴 shai
lcomparetw
otheoremso
fA. Zygmu
日dtyp
己 OllobsolutelyconvergentVilenkinFourierseries

L !
nitrllidu.
ction
C W.Onn
開 TeerandD.Waterman([1J)
gavetwoc
onceptso
fboundedneralizedbounded
onVilenkingroups Makingu
seo
fGBF，T.S.
and疫e.
proved([2J)athεoremof"~，typeonconvergent
andL Y，HVilenkinFourieτsεries. Weshallcomparethistheorem'vv
ithatheoremwhich1SobtainedbyGBFforBF.
園，2. NotaUonsandDefi.
nli
UcTUshat aLetG beaVilenkingroup，tgroup， Thent
Sadiscrete，hedualg
roupX oJG 1[3J)
thatthereexista
11sequence{fX anda
~O~o ofcharactersinX suchthatabeli
2nN.
叫。

)
(1iisofprimeorderム foreveryn~三1，(2)ト(3)X ニ U)( ，(4)号ヲn(5)=1foraIlx ε G，{Xo} ，札 r
her
e
1I/

εforever
yn孟 0，

εXn forev
ery1'1;;; 0。

ニヱLetrtl
u=1andrI1n=P
l・"九 Iorev
ery1'1詮 L I
fk孟 1andifk。
三五 j孟 S，thεnwε;0.

k h =￠ThenXn

二‑ 81。

五三12<with0壬めくあム forai111JJ二 O

1TheannihilatorofY.UNO82i8denotedbyGnforeverynミ;;U. ThenitISdeart
hatG= コ コ G2…，nGn= {O}n=Oandthatt
he{Gn}~，ニ o 18afundamentalsystemofn
eighborhoodsofzeroinG. Furthermore，for:eachn孟 othereexist
sanXn e宮x=:EGcanbεrepresenteduniquelyandeachxsuchthatw
ith 0壬んくん 1 foralljミ O. Thenwe

Jニ Ohaγe倍。)

fora=(ibx:MForeve
ry{九}j:l， 。話ん<戸川 1 (0妥 j壬幻‑1)フ Wεηl

明二玄 b，ぬZ

whereq=

j=O

j=O

Th
合nthecosetsofη
Gi
nG ar
e.
Gq，
n十 Gor 0三五q
孟 m，，‑1. Vleput
nf!iI:=トo三五 q~玉 mn- l， n=O，1，2，"'}.

DEFIN!
T!
ON1. Hfi
safunctiononG andifH cosc(f;H)=sup {1‑f(y)I:x，y el
1}

th合目

ωDEFINlTlON 2. AfUl1ctionf on G isofbou
i
1ded fluduation(BF)iftheree
xistsaucht
haiconstantM s00

2:孟M
mニ 1foreverγsequence{ 二 1ofpairwisesubsetsin~ .
DEFINlTION3. AIunctionfonG isofgeneralize
dboundedfh
必sup 之~ OSC(J~Zq， n 十仏.，)n孟 oq二日<∞

8DEFINITION4. I
ffi
safunctio
nonG，thenthen‑thmoduluso
f=sup{Iif

X，Y症

oIfi
s.
x‑yeGn}3. R側 w
tsT.S.QuekandL.Y.H.
([2J )showedthefollow
ingtheoremofA.
onabsolutelyconvergentVilenkinFourierseries.
THEOREMA.
afonG is GBFaηdijtypebsolutelyConvergentVilenkinFourierSeriesOnTheorems01A.ZygmundT)
少eonA

83

ヱ{九+1ω'n{j)}山<∞，(6)

thentheVilenkinFourierseries011isabsolutelyconvergent.
SinceBFcGBF，TheoremA impliesthefollow
ing:
THEOREMB. 1
1alunction1onG i
s0
1BFandず (6)holds，thent
heVilenkinFourier
11isabsolutelyconv
ergent.
series0InthisnoteweconsiderwhetherTheoremA includesproperlyTheoremB.
THEOREM1. Su伸osethatthereexistsη>0sucht
hatP
n+1孟 ηmn{n=O，l，2，…). lf1i
saβmctiononG andif{九+1ωn{j)};:'=0 isbounded，then1お 01BF.
PROOF. Let{Cn};:'=lbeasequenceo
fpairwisedisjointelementsin~. Then{C，l2，3，…} iswrittenasn :n={Zq.
川 +G;
孟k，n=O，l，2，"'}.
n:1孟jnThen∞ ∞ k "

玄ん/mn=~ ~ IZq"n十 GnI
壬 IGI=lButthereexistsM 孟 osucht
hat

Pn+1ωn{j)壬 M (n=O，l，2，…
)，andhencewehave∞∞ん∞玄関c，j(Cn)= ~ヱ osc(j， Zq"n+Gn ) 孟 ~ιωn{j)壬Mf1.
三五 M ~ kn/Pn+1~五 M17- 1 ~ kn/mnn=On=O

Thus1isofBF.
COROLLARY1upposet均t. Sucht
hat仏+1孟 仰n(n=O，hatthere.
esη>0s，，…).
12q.
e.
d:
ThenTheoremA iquiv
alenttseo TheoremB.
THEOREM 2. s:ゅρose that there exist a monotone decreasing sequence αnぽ'=0 01ρoat;sitiv
enumbersandasequence{んは~O 0nteger
swhichs
1nonnegativei
幼(7)ヱ(Pn+1αn)lf2<∞，(8)ヱkn/mn 1，
孟34Y.UNO

ぶR"内二∞，=01'
1∞す制(10)ーか)}く∞.

r=n.H

fonG sucht
hatfお ofGBFbu
lnoto
fBFandlneni
hereι
xist
saco
ntinuous
hotds.

oandv
z

PROOF.

Letqu

ここn作q

一一==1，2，3，.
.
.
)
.

Th
εロ o五q"五主q
71+九三五 mn for台v
ery17主主1We

11qll+弘L~;1==り(Zo. ，，+q=ql
官whereG~= tP

k. Thenn= on=，12，3，…
， ar
epair
wisedisjoint. Wedefineqll十 k，，‑lIa~，JPU)=={
白ifx 内 け¥j

幻ニ0，12，.
。
ら守=.
q
7
1ot
her
τiiiise.
Vv己 showt
hatt
hi8r
unctionfs
atisfiesth
econdusiono
fTheorem2.
lt!
sdearthat+ =0 (0孟 q三 玉 ‑トG‑nJ=αn玉三 q五三q，，‑，卜k，， ‑l).

IfV¥
Teput
二ITIi
11{f=主流ート 1 :k，手。} ，then
五三 t:l:N(n) (q，，+丸三五 q手 m，，‑1).

Thereforewehavei α i fknヰ 0，ωnV)==~1t:
I:N(n)

if，
k0.
nニ.

Thu8fisacontinuou
sfunctiononG and(6)holds∞ q"トk"‑l~:>~η=0 q=q"十Gn)=~τ=，ニtheassumption(7). SinceOnThe0
1官ms01A.ZygmundT
y戸e011 AbsolutelyCOr!v
ergentVile
ηI
11nFourierSc?
ies

8
5by(9)fi
snotofBF.
フNowweshallshowt
hatfi
sofGBF. Fixnミ 1. Foreverysミ 1，vvechoosethessuchthatintegerr…Pn+st1二五九ら‑
l<(qn+sイん+8) ρ(削 1Thenwehaveqnト kn~玉 r1 三玉?主孟 r3 ~五・・・三五 nZnJAJμtよaつρu﹃一一︑仏HU1出nvnf川 ) 恥FnCGSULetro=qn卜k.
Thenwehaven間"‑1~ osc，j(Zq.
n十qn十んl刑"‑)2; osc(，fZq，n十Gn
十2: osc(，fZq，n+)
q士山+ん

q二 q
"十.
Z2208CMZ
仰ート Gn)Sinc
eosc，j(Zq，η十 G
話 αn+sforq孟 r
1by(12)，wehave)nS‑
mnー 12:osc，j(Zq，n十 Gn
壬ムρ n十 ヱ)

九1)
αn+αn
玉=k+L:γk偽(九十 l…ρrt1+2:α8・nア二η+1s=lTherefor
efisofGBFbytheassumptions(7)and(10)
.
q，e.
d.

z
ψItosethatt
hereexists σ(0<σ<1) such that九十 1‑THEOREM 3. 5Thenthereexist{ αn}~=) andん)ご~O w
hichsatis)
ラ conduions(7)‑(10).

PROOF. Wechooseτwithσ<τ<1 andputMニzmrlp

n=Oαn=:m‑;;T

η=0，12，・.
.
，)(n→∞).

86Y.UNOん[m~/ M J η = 0，1，2，…)

where [xJ denotestheintegralpartoIx. Theniti
seasytoseethatthesesequences
7)
‑(10)
.
sati
sfytheconditions(q.
e.
d.
COROLLARY 2.
thattheree
xisお σ(0<σ<1)sucht
hat丸
斗 1 =0ThenTheoremA includesρroperlyTheoremB(n→∞)

Referel1ces[1J C.W.Onnew色erandD.Waterman，Uniformconv
εrgenceo
fFourierseriesongroups1，
MichiganMath.Joum，・ 18(1971)，265‑
273.
[2J T.S.Quek旦ndL.Y.H.Vap，FactorizationofL
ipschitzfunction
sandabsoluteconvergenceof
Vil巴nkin‑Fourierseri巴s，Monatsh.Math.，92(1981)，221‑
229ilenkin，Onaclasso
fcompleteorthonor
malsystems，Amer.Math.Soc.T
rans，.
l28(1963)，[3J N.J丘. V
1‑35