61.VI，No.2，pp.61‑64，May，1
959TheScienceReportsoftheKanazawaUniversity，Vol

NotesonGapTheorems
byNoboru MATSUYAMAり(ReceivedDecember2
6，1958)
L Introduction(:x) be an L~-integrable and periodicfunction with period1，and satisfythe
Letffollowing conditions，(1.1)

ff(x)dx=O，fン
ω2dxニ 1，and moreover letR(n) denote

ω=(flfω 一 山 )1dxyペ(1. 2)

R2

whereS，，(X) isthe Jt‑thpartial sumo
fthe Fourier serieso
ff(x).
1n my previouspaper 日] we proved the following twotheorems.

ffor anyα>0and fo
rany integr
al numberT.
ミ1，TheoremA. IR(JZ)= 0(1/(logl' ll)勺，(1. 3)
thenthe seriesゎムf(llkX)jk
一両面 10
瓦五二7σogpkY石 市L示方converges at almost every pointsi
n the interv
al(0，1)，where the sequence {lIdsatisfiesthat(1. 4)〔(i nKM，〉(ii) itis the sequenceo
fintegral numbers.

豆;
，

ffo
rany0く α
TheoremB.I(1.5)

R(1Z)=0(1/1Z")，

thenthe convergenceo
f2.
j akl2下FF(log均 2く∞，
1

implies thealmosteverywhereconvergenceo
f agap series(1. 6)
2.
jakf(1Zkxλwhere itsgapc
onditionsar
e givenby (1.4).
1n thisnote we discuss the almost everywher色 convergenceo
fthes
eries(1.6)
satis
fyin
gthegap conditions(1. 4)，and weprovetwotheorems，eacho
fwhichistheextensiono
f TheoremA and TheoremB respe
ctiv
ely，
Theorem1. I
ffo
ranyα>0and foranyintegralnumberT.と1，R(的 satisfies(1.3)，
then forthe almost ev
色rywhereconvergenceo
f(1.6)satisf
yingthe gapconditions骨) MathematicalInstitute，Facu
ltyo
fS
cie
nce，KanazawaUniv
er
sity

62

N. JVlATSUYAMA(1. 4)，itis su妊i
cientthat

2
J(1.7)

1klog止 log
2止.
.
.
.
.
{logpk)l‑2"(logp十 1k)
2く∞。

Th母orem2， Iffor anyα>0，R(n) satis
fies(1.5)，then for thealmostev邑rywhereconvergenceo
f(1.6)s
atisfyingthe gap conditions(1. 4)，itI
ssufficientt
hat:
s::+

ωif o
<a，then(1.8)

町

2J

止1/1'‑ (logk)
2く∞，呂忌1/2 (logk)宮く∞，

1

す<a，thif叩(1. 9)1

wheref
1=1+2α。

(1.10)For th世 proof of these theorems，we makeuseo
f the severalresults obtained in[1J. I
fforintegral numb邑rs2 and N ，2+1<71jく町三五2+ N，then it followst
hat
1(1".，.
̲，.
."
.
，̲
̲，.
.d.
.
1.
.
.
.
.
.
. 2 ¥2，JI
XI<Rlv:).
.
7(1. 11)from which(1. 12)会))C
立1ldK12f1
754k(1+NR2(2. ProofofTheorem1

I
fwe put in(1. 12)
2=S2 andλア=(s+1)
2‑s2=(2s+1)(=:=1¥その)，then‑fi‑) αO(s心)0叫l一一一2一j0I
κ心s〔οlog、p〆かr¥(logps
心一ト
J=司1+NR2可((Lλrト
J=1十f~VV~&I"'./)

Hence，fromthe convergenceo
f the series
国(2， 1)¥1/2 、/

1(8+1)'

2J 12J821LK=S2十)eogμ)"'(l7i
一一it follows t
hatthe almost everywhere convergencεof)(2.2lims→∞/r， =lakf(判的.

2.1)，we putf(To verifythetrutho
2tt(g(止〉ニ止 logklog
2止………(logpk)
2，l‑logP+1k)
then(8十 1)
' ¥∞)
'1/
2 ∞ 1
1/ /(3十 1/.， ，JEiZIU JJGK12j=JZ11弓JLF
し 川eバ∞ . ， /eL/(8十 0'二ニニ{刃:S::2J ogμ)" Vg<'(S2) ¥g'，十 rk:s自1(lfC!

¥1/2g(均十u¥
.
.
.
'''.
/)

nly.
rov
edihi
scasewasa
lreadypn[1]，soweconsiderthecase(a)o持) T¥
1/263λToteJ'0
11 Gat Theorems/∞¥1I，~ ! ∞く(ヤ一一二一一二一)lF
ゴバ logz'sY" g(Sりj(s+O'¥1/出g(的)/(~2jし田 1k=zS2+1

1. 7) and the above last inequality，weobtain (2.1) and from it the almost
From (everywhere convergenc色 of(2. 2) holds.
r
yto prove theconvergenceofseriesNow forourpurpose，wet(2. 3)

max 1 2j akf(nkくm く十1)
2Ik=S2十くS
2~ Is 叶~dxく ∞1a.
9However thegeneral term ofth己 S巴ries(2.3)isless than[!OgN(s)] [N(S)2 ] ρ X I 82+(.H+j)~!'日。(1)logN (心 2 j2j~v=l

I k=82十回 2'ト←u""Q.)[!ogNc，，)]Csト0'

v=lk=，<j 2十 1=O(l)logN(s) 2j2‑" N (、r)2j12akf(nlc.
xう1‑dx/02 ¥ 館 、¥山 ノ ノ!
1+2v (l Ogl)-~- )¥40427;川(2. 4)

Ontheotherhand the 1ast term Isless than¥[!ogN(.
，)
J(logps2N(s}‑1)‑2 =0(1) 記 ( logp

仁!ogN(，.
)
J/。(1) 2j巴;:;::;0(1)(logs)(logpS)‑2α@(2.5)

From(2. 4) and (2.5)itholds that thegener
叫 termofser
ies(2.

isle日sthanくs + 1 Y l o g
J0(1)k2Jla，，12slogSn~8'十 r'''1"'~b" (logpS)2α間(8+1)
logk̲̲̲"'，"，¥g.
.
tl)
2 g(k) 一一一一一~e-<O (1) 記三三
2J la/c1
同 2g(k)k=s?'十，， '..~./v必(l ogp.ト 1 k )2 -::'Y'..~./
十1の勾(2.6)

1Thisand thehypothesis (1.7)of Theorem 1deduce theproof of Theorem 1.

3. Proofof Theorem2
Ifweput in (1.12) byus告 ofμdefined by (1.10)，
Z=sft，and N=(s+l)I"‑s町三λ7ぐの)，wecan immediately verifyt
hat the convergenceof∞ /(8+1)'"¥1/2~ (I.
i la，，12 )' Iく∞A、k=s[.
t十(3.1)

implies thealmosteverywhereconvergenc日 off
8t(nlc.
x).
lim宮 内 f(3. 2)

8 →∞ん ~lHowever (3. 1) i
seasilyverifiεd，i.e.
∞/(8+1)'"、1/
2∞/(8‑トょ)'"¥2JI2j lakl2J' 訂(2Jαげ が /1，九(1略的L1Ju.
r，1"'，.
.
.
.L2))
μ(J1\k':'":'" 十 I~"I J ‑8"';;'1¥k::.~ 伴十 !U- I.:I¥.
.
.
lU 5 ，.
.
.
.
.
.
J k1logk¥)
2sロN.MATSUYAMA

64
1∞/(8+1)ド¥
1/2<::8一一一̲̲"2(::8 VVJCI k1 N (l略的 2 1's~l y S(lOgS)
2
し=汁十 I'.
.
.
.
.
""
6I'V/ )

f∞¥
1/2 1 ∞ い +1)
い¥1/2~I Lj一一二一:.)'(::82Jがか(l略的2 十¥(lOgS)2 I .
¥8"'‑::1k二件十 i t.~fel

S豆 町1〉

"‑.
1
1‑"‑
"
5'""'.
‑/)(811GK￨2が 匹 〔 知yyl2fIn t
h母 case (α)，evidently the above series converges，and then we obtaintheconvergenceo
f.2)。

Lastly fort
he completiono
f theproofo
f Theorem2，iti
ssu
ff
icientt
oprovethat::8I.3)8=1((The general term ofmax

ct/c8t2，くm く CS+1)tL Iドf川 川 作 く ∞.3) isless than[logN(S)][N(8)2‑'] ('1 I 81'+ω+1)2"::
80(1) log N(s)S冒'l~=oo'i2

訂 向I 1

1 dx

Ik=s[.t‑j‑吋 U十[10gN(8)] [N(8)2‑'J (8+1)μ:8‑::8<0(1) logN(J');(，'+1)
ド三玉(::8(3.4)

官 ~1U~O¥¥J!
log N(s)JN(s) ::8、k=stJ.十 、

ak12 (1 +2V(2~+川 2 叩〉

1k~8 tL +lJ2‑吃 (2叶 1)"，，‑2"tL

v=1Sincei
nthe lastterm，itholds that2"(2α 十 1)2‑2<'1'=λl(S)(2 叶I)S.-2叩 =s(加十 1)( ト 1)-2~1' =0(1)，(3.4)isless than
(3 .
5)

(8+1)

0(1)::81

2"∞(SCト1)"

N(J、)logN(s)記 2‑ .
s;
:
Vk=sT
J.
ー ト ? !
=1

2k1‑1/1' log止!
c=s!
J‑十1

Now i
f
α 呂atisfiesthecase (σ) then6)(3.5) and (3.

1 ̲10く α三三つ) a.
nd 1一一一一三二μ一 戸

中indicatetheconvergenceo
f@and we complete the proof ofTheorem 1.
Referen間[1] N.Matsuyama，Ontheconv
ergenceofSorneGapSeries，TheScienceReportsofKanazawa
UniversityVoLVI，Nomber2，1958