Sci.Rep.KanazawaUllIV.
Vol.8，No.2，pp.2
71‑277Septernber1963

A Note on Convergence of Gap Series

NoboruMATSUYAMADetartmen
tofM athematics，FamltyofScience，KanazawaU削 versity.

(Receiv
ed28Jun
巴 1963)

1. Introduction. Inthepresentnote weshallassume that!(t) isanintegrable
functionwithperiod 1，suchthat~ ! ( 何 t= 1，~ !(t) dt= 0，{n九叫dis a sequ児問e白nceo
fpositiv
e日nt巴gerss
atisfyi
ng叫
nk
ト‑刊￨
andμK
匂ac，R. SalemandA. Zygmund [1]provedt
he followingtheorem;

f， forrt>J，!(t) satisfies
TheoremA. I〈ザ( 1 1 ) ( ims(t)￨asn→∞， then(1.2)

L.
.
.
: Ck!(n"t)

convergesa.e providedt
hat勺

(1.
3)

L.
.
.
: c~ (logk)2 く∞，

whereSバt) meansth巴 か thpartialsum of theFourierseriesoff(t). Ontheother
hand if，for0く αs.
:1，!(t) satisfies(1.1)， then
(1.2')

ェ」
万一 !(nlct)kμconvergesa.e.，provided t
hat(1.3')

一
三一s>lInTheorem 1andTheorem2oft
his note，wesh旦11prove that(1.
3) and (1.
3').
may be rep
lace
d by more weakerr
estr
ict
i
onsrespectively. Onth巴 otherhandSIzumi [2]obtained a result concerning thespecial case β 1 . Our Theorem 2巴sult.
(iii) contains ageneralizationoft
he Izumi'sr‑271‑

N， MATSUYAMA

2722， Theorem1， If， for a>l，f(t) satisfies(1.1) and for som色 noト negativeinteger σと 0，L:

(2.1)

(logσ 十 jk)2 く ∞holds，then (1.
2) convergesa.e" where loga means the a times iterationoft
helogarithm
ofoft
histheorem，we need twolemmas，For the p1'oLemma1，
H，forα>0，f(t) satis
fies(1.
1)， then fo1' anypos
itivei
ntegersM
andN suchasM く N ，and fo
rany 1
10n decreasingsequence{r(k)} ofi
ntegers， we岨Nl'u¥1jJd2︐TLV3PJ/{¥mvkκ'τノQUt¥j/1¥mkf之aLrj

ck

NZM‑‑(2.
2)

Iohave)‑u.

N;a) L:c~!
M¥fwhere F(M，N;α) isdefinedby

サ for 。くい，F(M，N;かO((N‑M)l= O(log (N‑M))for a=し(2.
3)1)

fo1' α>1，

Proof. I
f we put， for simplicity of writing， 'Pバ t)=j(t)‑S (t)， then itisη

familia1' thatforjくムふり)(幻'Pて川Now fromthe above1'elョtion，wehave1(NN[戸川 τ川 nkf)fldtニエ;
C~1M

1M0'1¥ 'PT(k)(n川 dt01+L: CkCj¥CPτ(k)(nkf) CPr(j)(幻kキ J J ON、N‑l N¥

N‑l N‑j¥23cfoliogT(h)j‑24+2A23JKCJOL(h j〉
一 α(logr(k))一αjNニエ ct01logT(k)卜α
2十 2L: L: Cj+ CjO(r一日(logτ、γM¥α︑︑﹄﹃Fノi3j'J¥SJJ〆Jt︑︑K︑¥Ub

Tol

N‑M 1ur1r)‑α2 +O(1) L: 一三円pu¥Jおり hE曹Irlogrトーr))=11'ff¥

Nく O(1)L: c~M

j=M

ニニ

Hence we obtainthe lemm乱.
Lemma2. Unde1' thehypotheses of Lemm昌 1，we have
1(2.
4)

r~ IN

12N¥lL:ckSτ(k)(nkt)‑
1dt;:;玉F(M，N;α) L: c~.
JOI M

M

The Proof issimila1' asthatof Lemma 1.
placeCPτω (nkt) by {5N(k) (nkt)‑5'I(k)，}Remark. 1nLemma 1，we may 1'eJ
where M(k) andN(k) are both nor
ト decreasings
巴quence ofintegers suchas M(k)A NoteonConvergence01GapSeries273<N(k)，and we have the following inequality，;;
I5cdS
N(川ProofofTheor・em1. Putting，fork=l，2，.
.
.
，(2.
5)

ToCk)=[exp Ao(k)J and ，10 (k)=1/k ，thenwe have1

L:Iの 110MTO〔川kt)Idロo((logTo(k)) )αICIc!

ーく∞Thisformula indicatesthe almost everywhereconvergence of(2.
6)

L:Ck伊τo(/c) (nkt)，Now putting，fo
rq=1，2，3，・・(2.
7)

Tq(k)=[expん(k)] and Ag(k)=(logg k)2/α，then for some ko，.
Jq(k) isn
01
ト decreasingas k己ん =ho(q)，and wehave，for巴ach
q=O，1，2，・・・， thealmost everywhere convergenceof(2.
8)

L: Ck{STqC/，)(nkt)‑SτHl(/c)(nkt)}.

Forthe proof ofthis，weconsider asequence of trigonometricpolynomialsνk+l‑1(2.
9)

Xk(ト JLCI{Sτq(l)(nlt)‑Sτ
肝 l(l)(nzf)

where{vd isanincreasingsequenceo
fintegerssuch that，forko(q)(2 10)
園

VL=somec
onstantA，k++1=V〕
]
，Vk
1=V
k十+[‑2‑MhL10gc

Since thelargest frequency ofXk‑1(t)is rg(νρ n(v/c) andthesmall巴stfrequ巴ncyo
fXk+l(t) is T引ー 1(九十 1) n(ν廿 1)，andmoreover we have fromt
he definitionof{ν k}(2.
11)

τq+1(νk十 1) n(νk+l) >e>l
一一Tg(νk) n(νk)

た ko)，1(wh巴ren(k) meansnc/(h=1，2，…
)， sobothsequences {Xt);2k乙ho}and {Xt);2k‑2k(2k‑1とれ}are orthogonalr
espectiv
ely，andby Remarkof Lemma 1，11tの川)川￨
円
叩t=L~ I
L:Xれ2バk(d:~ IX
似叫(げωtの仰)川げ∞Jo0ν2k+1‑1:
三~L: ̲L

1=ν 2kcr(logτq 1 (l))ー αく Zc?く ∞ .
十From this and (2.
11)，エ X2k(t) is the Fourier serier of some square integrable
functionand itconv巴rgesa.
e.
. By the same way，we obtain theal
most everywhereconvergence ofL:'YU‑l(t)anc
lconsequently L:，Xι(t). Now the convergence of (2.8)
followsfrom
1(2.
12)

mmax
.
1~ c/{Sい )(nlt)‑STq+l(l)(nlわ}¥2dt<∞Joh〈 附 くνk+l 11=1ノkL:¥ι274N.MATSUYAMA

But this(2.
12)iseasilyv
erif
iedbyuse oftheM引 lchov'swellknowndevice&. 主b，E'~lef
thand sideo
f(2.
12)i
sles
sthanby (2.
7)

ヰ (log ( 山 ‑Vk)Y ~ol 日) 2dt
1

イ:=;
:
O(1)子(logAq(ννHl‑1(FJE
V
K刊日(l)一 αllog

三五 O(l〉「(C¥

く O(1)
L;c~Asaconclusionofthe above arguments，we obtainthe following山U山n凶n吋
1d
由山
附e
い(2.
13)

討itiv
ei
nt巴ger，th巴剖叩
11 L;
posk=kOcn

九Ck{f(1幻九1糾一Sτ
弓q(ゆhめ)(幻叫I

converges旦.
e.
Lastly if weprove，forsomeq，theconvergenc巴 of(2，14)

2Ck)

thenwe obtain Theorem 1. UsingL色mma2 inst四 d of Lemm旦 1， we can 色asilyl
ead，thesimilar methods as the above arguments，foreach q=O，1，2，'"thealmost everywh巴reconvergence ofνkト1‑1; CZSTq(!)
ユ咽L
1(2.
15)

t=νkand，have，foreachq=O，1，2，'停

11押max+，=:凸If: CzSTq(I)(幻 zt)

Joνkく 附 く 勺 +1 11=ν

三子 (log ( 山 h))
2
10lEh仏 ゆ1 υ ..
"‑1'(2.
16)r
工作

ヰ (lOgA (Vk)q

O(1)ザ(10叫)2

From (2.
1)，thelastser
iesconverges whεnq=σ. Thusby (2.
13)，(2.
15) and (2.
16)
we obtainthe theorem.

;
3. Theorem宮。If…
一satisfies(1.1)，thenwe have，(川iυ) 伽f
O町r 0
く<αくK
÷ a凶nd 凶s>1
ト一 α，(け1.的2勺F)c作〉
÷十， (υl2')山∞…v

凶dβ::;
aく<1 川(ιめi
川りi) 伽f
O町r ;三an川〉;

川

f U = I andF

C011nkf)z も/忌f:((logk)s
一convergesa.e.

ー

275

A NoteonCo回v
eγg
enceofGapSeries

Proofof (i). Ifct and P satisfy the hypothesis of (i)， then we have for
eachq=O，1，2，F ¥ β 2 αfq̲ a'Q+l
一一一一1‑ct.
‑‑r
‑
1‑a'
山 寸

where ct'=(1‑ct)jct so we determine anumberr such asfo
reachq=O，1，2γ.
，o(where σisanarbitrar
y but fixednumber)(3.1)

32‑ ct，q‑ct'
Q+l:
J 一一一一つ百工ヲ一一‑.‑'‑̲.">r>f
ハr >1，1‑ct

1ー ぴム

and we dcfinefo
rq=O，1，2，'''，0(3.2)/，
Ao=kcr‑s)ja，Aqlz)= k(r‑2β)/α Aq(k)UTq(k)=[expAq(k)J.
十 1(From(3.2)
T‑2s 1一 日Iq

ー 一 て ， ‑ 一 二7 十Aq(k)=k山穐Tー s二了
一一山。

α，，.

山

holds，and fo
reachq， ん(k) isnor
トdecreasing as k tends to +∞， and for eachk >1，and q=O，1，2，… ，0 ‑ 1，AQ(k)ミん十l(k)，Evidently from (3.
2)
1I::‑‑;
73~ ITo (nrt)Idt
砂。
Y'く∞(l)follows，andt
his shows the almost everywhereconvergence of(3.
3)

デ 古 川 げ 印 刷t)}

Usingthe same formulaeas(2.
9)，(2.
10) and (2.
11)，then fromLemma 2， wehave1:
ipxzk(巾=干 1:
1hk(t〉

ν2k+l
‑1 1

<I:
: ~~2i ‑‑‑‑‑‑;;0ーで::‑ (ν2k l‑V2k)
1ー α
一l=ν2
k['"μ，
1q+1 (l)U十三I::Aq(l)
1ー αν 2k十~-1I:: 一一一一一一一一一I=~ 一一く∞1=
ν2k[2βZ
T‑2sAq(l)UU'ZTThisshowsthe almost everywhere convergence ofL.X2kCt)， and by the similar
way，wehave theconvergenceo
fL.X2t) andc
onsequently L.X1，(t).
k十 1(Nowwes
hall prove fo
reachq=O，1，2，"'， 0.
m

1

(3.
4)

L
.¥
kエ 告‑{S川maxJO 勺 く 附 く 勺 +1I1=
νk1
μSTa+1(l)(nzt)}‑
1dt<∞)(n，t)‑ョholds. For theleft hand sideof(3.
4)isless thanlog(νk+lー νρ ν(k+lー νk)2
三戸 log(ν".‑Vk)k κ γムv=O
u=Oエ→，エ一旬

j1lh市 十 1)2υ 1つ月1‑{{SSTTqq(([[))((nnzzt)‑ST川l )(nzt)}t)一品(nzt)}r‑dt01

円ー/=νk十 u2り

iq+ ((1)Iμι￨N.M ATSUYAMA

2
760ー1JT‑i72U(1ーαk十 (21〕25 2logAg(h)I
勾 YK)(ν吉一νρ 2‑V ν〕
u=ov=oAq十 l(l)U

1=九 十 叫 [ 叩ー1log[logAq(l.q(l)l).
/~~Iα =，.
;;'‑
1，~一一一く∞
三三~ ~~~~.~;.， /ααt
' [2s[r‑2sAq(l)[r3.
Thusby (3.
3) and (btainthaat4) weo(3.
5)

7L六品仰戸t)lconvergera.e.
epeat near2.
15) and (2.
16)，we can
1y t
Now ifwer
he same estimations as(concludethat thealmosteverywhereconvergenceoffy(3.
6)znzt)〕(holds，providedthat
Iog[z(3.
7)

r‑2s 1ー α r σ γ ‑s[2βJ 一 (1 ーの(~一寸土T+~∞σFortheproofofthis，puttingνk+l‑1 1zーも‑S"̲Cll(nzi).
…

Xk(t)

口

pl=vk [0andJl12h+[‑2someconstantA，
logー
cん (Jlk)]， 1
口thenwehave
11ωt
為幻以め仰)
2k(〆守干X
1L0l

2!
'.
:':
1‑1l̲n νr‑‑‑~(\ l- rt. ν 2 k+1一 1 1AOいσ(ωんk匂加川叫心り吋)
リ…
丙三勾
干卯
[2s一一之三 F
三忍A 子

一→α巴守写 ( 山 一 ωAlす σCl)1=O(1) L:ー αOnthe otherhand，wehave
1~ ¥J

押事

max'n咽n1t)‑
1dt，
L:‑;
主‑S"σ(l)(Iovk<m<九+1 Il=vk [ ' υ l̲I1(vk+(U 1)2 ー 1 1. ¥.
.
t.
‑dtn1t)IL:
1"1(め(つを‑S"(log(ν k+1ーν(k+1‑νk)2k)ν
.
.
，Jlk+lk)‑':og(t‑.
1 ‑J三平 l

応 。=0u=o旬rJol十V

l=vk十 U2Vr

<2:::logAσ(Jlk)L.σ(Jlk)(ω‑h)1‑P51Jrogl:L:L:!.o 2v(1‑a)= L:l
り 協i[2βlzhk

一 ー (J 1̲1arr

!
J())
(1 α
守主号，+う~al 1716(l)1‑alogん (=zYJrl)=
L: 吉1hzLI(l<‑Pーl=vkμk 1=νkA Note側 Converge
1
仰27701G仲 Series

These estimations inclicate that (3.
7) implies the convergence of(3.
6).

Ifwe take a sufficiently large (J， then by (3.
1)
(3.
8)

r‑2s 1‑ a，a r‑s2s‑(1‑a)(!

ーァ寸一一一l‑aα ) >1，consequently (3.
7)holcls，ancl(3.
6) converges a.e.
. From (3.
5) and (3.
6) the proofof (i) holcls.
Proofof 仙For a ancl s satisfying the hypotheses of仇キすbutaト) we

34LI'Lnrr >1，s十一(3.
9)

i‑ lp〉

ancl we clefine，
九十1(k)=Ao(k)αパ+1

Ao(k) krj
2α二(3.
10)

Tq(k)=[exp Aq(k)].

The methocls of the proof of(ii)are similar astha
tof (i)，sowe writeonlys，conclusions，that i[A] for anyq=O，1，2，…
(3.11)

L; ~(f(nd)-Sτq(l) (nd)}=1Zsconv巴rgesa.
e.，proviclecl 2s >1，[B] foranyq=0，1，2・，

(3.
12)

三十一品q(川tconvergesa.
e.，proviclecl(3.
13)

2βr
̲"q 11f:J‑‑;;‑αノ>1
十2

Since 0くぴ'く 1.
holclsin our case， forsu
fficientlyl
arge q，(3.
13)i
ssatisf
ieclandwe obtain the proof of(ii).
Proofof (iii). I
fwe put r>2s，Ao(k)=krj2，A1(k)=(log k)r‑2s(log k)
3， A2(k)=(logk/‑2s(log Jog k)3， anclτ q(k)= [exp Aq(k)](q=O，1，2)， then we can also

verify the v
aliclityof(iii).
References(1) M.Kac，R.SalemandA.Zygmund，A GapTheorem，Trans.Amer.Math.Soc.，63(1948).
heStrongLawo
fL
argeNumbersandGapS
eri巴s，TohokuM
ath.Jou，.
r3(1951).
(2) S.Izumi，Ont

11*) Forthespecialcase aニ ‑2‑andβ> 瓦， weput
ε 豆 (2s‑1)，thenthereexists，foragiv
enr>1，apositiv
eintegerσsuch that aε<rくσ( ト1)
ε.l
fw
ede
fi
ne Aq(k)=kTεq，Tq(k)=CexpAq(k)Jfor q=O，1，"…
・円 thenweobtaintheresultbythesimilarwayasthegeneralcase.