Sd.R
むIl.Kanaza.waUniv.
.29l'.
Jo.2pp.97‑
102VolD邑cember1984

AbsoluteC(mVerg:ence

VilenkinFourie
r:YoshikazuUJ¥TO。
1/Mathematics，

Ka符azaw
aUnillersitv(Receiv
εdOctober30，1984)
Ab自tnu~t 圃 We shallgiv
eatheoremontheabsolutecunγ日rgenc日ofVilenkinFourierri
εsbyusingt
henotionofλbound巴dρ‑fluc
i:uationS己1. Introdu
ctioJ11M.S
hiba([4J )generalize
datheoremo
fM.3ndS.Izumifort
hea
bsolutecOl
1vergencetionofλboundedv
ariation. M.SchramrnoftrigonometricFouriεrseries u
singt
hel1o.Waterman([3J )extend吋 theresultofM.S
hiba. 1nthispaper，weconsider
andDanalogueo
ft
heser
esult
stoVilεnkinFour
ierseries.

2. Notation
samlDefini.
ti.
0J¥'18Through thi
spaper，G denotes a ViIenkin group，thatis，an infinite，compact，metrizable，zero‑dime立sionalabεIiangroup. Thent
hedualgrou
pX ofG isaninfinite，discrete，countable，torsionabeliangroup. N. Ja.Vilen
kinprovedthatthereexist an
二ooffinitesubgroupsofX andasequenceρn}L10fincreasingsequencenumbersw
itht
hefollowingproperties:) )‑‑1( (二(Xo) ，Xo=1onthegroupisoforder九 foreveryn主 1，

X = UXn‑

LetGnbet
heann
ihilatorof '1orev
ery自 O. Then{二。 isadecreasingsequencesaf
undamentalbasiso
fneighbour
hoodsof0inG.
ofopensubgroupsofG andiLetn
ゐ=1andmn=ム…βnforev
eryn~ 1. Wetakε 'Pn己 十 l¥forev
erynミO. Foranintegerk詮 1，wecanr
epresentkas

k=2:ajJnj，O豆 aj<戸'J+l (0三五j三五as宇 O‑ 97‑YoshikazuUNO

98rpr

ThenweputXk=IIWehav
eXn= {xk:0計 <mn} forev
eryは O.
三V(G):~e denotethek‑thFouriercoefficien
tof1byForIE!(k)=fcI(x)瓦向 dx

foreveryk孟 oandwedefinetheVilen
kinFourierserieso
f1byS[jJ=玄/偽)ル(x)(xεG).

DEFINITION1. ForIEU(G)，0<ρ三五∞ andanintegerN，GO，
thei
ntegralmoduluso
for1i
sdefinedby
continuityoforderN f
ωIp(j;N)=sup {1β ‑/11p: yεGN}

wherejy(x)=/(x‑y)foreveryx，yEG.
DEFINITION2. Forafunction1onG andasubsetH ofG，theoscillationof1onHisdefinedbyosc(j;H)=sup {1I(x)‑1(Y)1:x，yEH} .

}Foreveryn孟 0，{Zq.
n};:~1 isaseqlienceofelement
sinG suchthat{z +Gと;
lisn
引nG.
allcosetsofGn iLetAbethesetofincreasingsequencesofpositivenumbers.
DEFINTION3. Forλ= λ{n}:=1εA，O<ρ<∞ andafunction1onG，wedefine
附2Mv~) (j)=supSUPmーl
N 孟o {qJ仙二。

び，Zι N+GN)}P{oscλi+l;:0‑runsover allpermutationsof{O，1，2， ，mN‑1}

where{qj}1・・If V~) (f)<∞， wecall1afunctionofgeneralizedλboundedp‑
fluctuation WeG)thesetofallfunctionsofg
eneralizedλ‑boundedρ‑
fluctuation.
denot
ebyGλBFω (3. Analogueo
fatheoremofM.S
hibaimdatheoremo
fM.SchramnandD.
Waterman.
Nヱck=landaミぬ詮…孟 aN，then

LEMMA1([3J.
) 1
1Cl孟らミ…ミ CN>O，

LM

Pしα品係

く=GNZMNl一N何ヱLEMMA2.

k=1o

LetN ミ andn孟 mN・ Thenん ￨ 仰 ) 一 11
ゆ =21命 ￨1AおοlutcConv
e得 制ι ofVilc1'l.
!
an‑FOla'1~er Se'F
iesnrzPROOF. Sincev
ehave

九kェ0

1 2のこ
11(ゐlLニ

2
1wherc ちv
ilecttartleteristi
cfund:
ioIloft
11εsetJistindD.
of乱 theoremofM.SchrammeTHEORElv
1二s==‑l
J1孟ρ<2randλニ十]ー/aBorel(λn} lEA.

().
.
'J，(m'NN=Othe刀 tonv
e1geseries fcheVile刀kouriersinF
PROOF. ForN 孟 O.weset。
Nニandl

句、ー=.
;
.
， 一
1nんニ 1. ;o..

ForyEwehave+y)

話叶 y)

1/
1
<~ (111/1')
Sdx}1/3(2‑T(トタ)‑fエ) l'=:::::;
UJ.lN

1ThusweG1)ーエ十 y

1l/r;
;行ry0
三1nforev
N‑l呂ndyE
玉q十2:q，N)q，Nー ト ‑ 寸 Z

孟{ん￨十ZIJ，N 十12dxdx}l/r)[PZq，Nand80，}，!
1.N‑1Zl
んはい十 Zq，N 十y)‑τγ:トZq，N) 12 d'
1=0

間

毛N‑1~

I十Zq，N'
1‑，
Zq，N) 11dxq=Oileseqtm
v
eryzEGaMeveryyEGJNpwet
Foreake{qJ1111Miliflatt{1f(x十乙'h，N 十 y) f，(H411N)￨}21i
smonotonεdecreasing. Lemma1impliesthat告9YoshikazuUNO

100n1N‑J2'+九九r+y)‑f(x 十九• N)IP111川 1V‑:E l

豆 m1Vtfl.
N‑l+Zq.，

十ZIL，.iV十 y)IP

μNλ;+1Jニ JI(oscげ}，X~卜 Z(jJ ， N 斗} P
λj+lZ
μN j=O
手附N

94十〆Fdfすxv

淘J Q

λーベlfJxf川mrz

Itfollovv
sthatがq=O話。 NmN14lvf)(f)

Butwe

foryEん1and0三五 q三 mN‑l，

卜Zq. 十川‑+Zq，

I2此 二 ι 1∞す吋

一一‑112= ~叫 1 21

1 2 1

11=0

11二‑112.

mNforyεThuswe

21mN { Z 1f(η)1112
γn=mlvmN‑1=玄q。
二γr;1んl手 y~t) (f)Ynj¥jandhence∞
Y4

1 2 1‑112壬vf)(f)lI
71ニ nl.^lThusbvLemma3wehav
ε2

2: Ij(n)1

幻二2

1n
l'
y

1‑1
豆ん F12N￨fW)￨21xn(yト 112のV?
)ぴ)1/rμN

Thεreforeweobtain十y)一角)12政

AbsoluteConvelgenceo
fT/ile珂kinFourieγSeri
お:Z l

玄~n=l

Nニ

壬If(η)
l;;.
z=mlvz1η)12}1!
2(WI
J
V十l一
1
1士 川N

m
Z
K

λl=O

101

ρ
:
JJ
27く∞.

ハu

話(1/;町内)

q.e.d.

COROLLARYl(Analoqueo
fatheiコ
remo
flVLShiba). LetJ話予く COp :l/r+I/5ヱ 1，1三玉ρ
21・andλニfi
ηGλBPが (G)aBore!

{ん)二 1E三人

zd41mr λ
Jにω00

山ω {j'，'(2‑P)←2T<∞?

P/Nニ Othent
heVilen
kinFourierSeJゴesfi
sconverges

PROOF. Sinceλ n}~=l ismonoton

wehave1¥盟百十ζ二 、kニ 1A/<‑‑" NandthenC
j.e.d.

1followsIromTheO
l・εm.

COROLLARY2BF<l)(G)and~{TN+lofatheorer
nofA.Zygmund [2J).Il二二G {1};N)}山<∞?

N=Ofconvel
努esα
ThisCorollar
yisthecaseofil"ニ 1，ρ=，1 r=1inthentheVilenkinFourierseries
圃

COROLLARY3([lJ). LetG be.
ie.，supρn<∞は(ιんλn} IεAα
仰幻 df
εLi
匂pα 什GλBF(1)刊(G)
. 1
1んλmN

二F01i
げ削
幻Y1;li;εersωεnたω6s，1.

Let0<β<α=
壬玉1，守 λニ

fc0，'η削1ωμ1
仰vet
培
2ぎZ♂
釘S

PROOF. Sinceヱバート 1λ えω∞(f;N)1乍 mst A
N=OαコthisCorollar
yfoHowsbya
ppl
ying

CO

(βα)/2~"-<∞，

1as1
う=r=1.
C
j.e.d.

E
加feren.ces[1]C.羽T.Onnew
eer，Absolu
teconvergenceofFouri巴rsεriesoncertaingroups，DukeMath.]
ourn.，:
1宮

YoshikazuUNO

102(1972)，599‑
609.

[2]T.S.QuekandL.

H.Yap，FactorizationofL
ipschitzfunctio
口sandabsoluteco
即 日'genceoi

Vilenkin‑
Fouriers
己ries，Monatsh.Math.，92(1981)，221‑
229.
[3]JV
1.SchrammandD.Waterman，Absoluteconv
ergenceo
fFouriers
巴riesoffun.
ctionsofA
BV(ρ)2.nd，ActaTv
latl
LA
cad.S
ci.Hunga，.
r 4W(1982)，273‑
276.

[4] M.
.S
hiba，On absolu
teconvergenceo
fFourierse:rieso
If
1inctionsofc
lass A‑BV(，)
tS
ci.Rep.
FukushimaUniv.，加 (1980)，7‑10