Matsuyama N

Matsuyama Noboru

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1 The Science Reports of the Kanazawa University， Vol. VI， No. 1， pp. 1-7， April， 1958 On the Convergence of Some Gap Series By Noboru MATSUYAMA (Received September 15， 1957) 1. Introduction Let f(x) be an D iー此 egrableand periodic function with period 1， and satisfy the following conditions， (1.1) fiωdx=O， (1.2) fi2ωdx=l The Fourier series of f(x) and the砂 thp呂rtialsum of it are respectively， (1.3) ???????? ? …????????and ?????????? ? ?? ??、 、 ，??? ???呂ndR(n) denot日S(1.4) Rω=(Jlf( In this note we shall prove the convergence of a gap series (1.5) Izd万/同〉under some restrictions on R(n)， where {nd is a sequence of positive integral numbers satisfying (1.6) 。く111くn2く…・・・く舟kく・・・…，and Lp(止)is defined for arbitrary numbers α>0，戸>1 and a non司negative integral number t， by the following I Lo(X)=x1- α (logx)~ ， (1.7) 1Ll(X)=X(logx)!ーα(lObX)F>， \Ll{.X)=X(logX)・…・・ (lo，gP-tX)(!Oれり1→(lO，S'P+IX)~ (t孟2)。In the sequel we suppose that logox means x. Theorem 1. If for any α>0 and an integral number Tミ1，f(り satisfi巴S0・8) R(n)=O(~~ì ¥ (logp理〉αjthen for almost all x， (1.5) converges， where {JZk} 1s a sequence of (1.6). 2 N幽 M.ATSUYAMATh記o:rem:t If for 0.> ; ，ω 開制部R(却)=and {lZ，} satisfies (1 then the converg日nceof (1. 2:.i implies the ahnost everywhere convergence of .10) ??where s>l and T is a int母ger.TheoreJl¥l 3. H for any α>0 and an t = 1， satisiies UtU削げよ2，or for any いく;and(1.11) Rω= 百二戸)sthen for alJ工lOstall x， (1. 国 lLp-1(のconverges， where {1Zk} is a sequence of integral numbers 主主土L二三6>1対k(止=1，2，'0'The caseいく 1 叫 t=lof川 orem3， was p一dby M 恥， R. Sa討l色一l[lJエ， and als 心 by S.弘 Izu 立mi[2]エ， who 邑問e 加f臼m幻1一ε討thod1S 呂 the d出if妊E号rerロ1t way frol白}工n[目1 ι ther臼n.もV巴 nm羽w another of it仁固 S. Izuim 仁2] 司明wλliTeakerresults of the cases T = 2 and 3 of Theorem 3 than (L ー Theorem 1 is proγed use of the methods of Jo L. K.oksfua [3]， which treated the law of large 11umb邑rsof some sequencθof 3nd the of Th位。1"告~工1 3 i8 to that 01 Tb記orem1. It 13 inter邑 thatwe discuss the almost everywhεre conv母rgenceof (1.10) use of {lZ，} which s昌tis長esa stronger condition than .6). We havεtheorem below i1 this sense that i8 Httle g色neral1zedone of the theorem obtained by M. Kac [5J and S. 1zm.1! [2J。Theorel11 4. lf for any α>0， satisf記号 (1宮 and{n!c} satisfies (1.く∞，where A 1S an 乱rbitrarvDositi vθ 乱un1b色r.then from th巴 C011veI・genceof ∞ どZJb29 the almost convergence of ⑤10) IoHows. 2. PI'oof of Theo:rem 1 For the proof of Theorem 1 we need a lemma. Lemma 1. If and い/c} the 吉田 ofTheorem 1， then 101" numbers z(l~z) and l\1(1~l\1入 it hold8 that 01/ the COllZJt'rぷenceピゾ SmJlC Gap Series 3 (2.1) Jl 叶~V 1 U ，'d? J ，，/ N "r N2 R(z川~)一一γf(llkX)i2d:X~-T ~;_'2+0(.::'一一一 一jJ ".:言十1-.Lp(kJIVlkXxa:x二込乙ρ(402 L LP〔z〕2 / Proof If the greatest common divisor of n" and 1lj is d， thεn there 告玄isttwo numbers刀，'and Il/ such a色町 )=d，(地k，' 121/)ニ 1，7lk=d1lk' and 11j=d沼 7F.lf we suppos邑 thatthen and zく 1Ziくnι三玉λT+Z，dく(n./-niつd出向-1!j豆.N'，z 〆却ノN"-7=nj <.. from which it follows that by the Parseval's relation， jIfchxWAlzJム C'CtI 12121んりん(2，2) ;aR(n i ':めが)<R(合yFrom the monotonity of 忌)and (2.2)， itholds that f1 叶 Nlk宿主1五窃六万o-ト!" 1 {'1. ，_ 1 ("1 =ヤー:"'¥2-I f(llk:rアどh:ムヤ 一一¥-I f(nkx)f(iliX)d:τ J十1Lμノf lりJ ん〔めん〔j〉j月T ( _ ¥2 [ "+N ¥2 くー ニー-_，2 十R(<rI1記 一二一一i三 Lp(z)2 'LL ¥ N J ¥ 十1Lr{k) } ζ (N2R(z/N):.) 一一一_，2一+0(----'~~:c...._ 一的リノ ¥， Ll'(Z)2 } Thus we obtain Lemma 1. We now prove the theorem. In the inequality ，1)， ifwe put z=A" and iV=o.ートl)2-，F=2.l +1(三 λ110))，then r 叶~V 1 _ N，ω(N(A〕2)T一一j(河川2dx~一一寸\2 十 01 一 j ohZ11f，p(止).1'.."a" -' I ""=Oo Lp().ノ ¥LpO.z)zくfogpPλVo.)-l)2α/=o(同 gVog2A-:ゴ石A(l長司〉下ア，-) Thi邑 containsthe almost everywh日reconvergence of the series ∞ fλ十[)2(2.3) J51kl-ITP7/〔仰〉Whence it foHows the almostεverywh巴reconvergence of (1.5)， provided that for almost all x N. MATSUYAMA 伐王立川1〉2lA+1寸万 1=0For the proof of (2.4) we us邑 thedevice of P. ErdむS[4]. and (2.4) 4 For ev邑ryu=O，1，2，・ υ …9 2v， 日 Z ，13L閉めand f A' +(2，+1)1¥1(λb-' ~ mtdsil kEL+仰向門1五むず 1ミ一品-)~，4ρ1λ+(21G十l)N(À)Z-'4争jiJ+…〉 Lp(k)f(nkx) have we A2甲吋 4 ¥ [，J21ogA"'(logpA)1-O:(logP+1A)sJ2 J A2 2-2V v4 ¥ [l2!ogA logzA…(logpA)1→ 門)/ 引勾_V¥ ~f v42-2V ¥ ~Ol ニÃ3--)+ 0(~p)Whence it follows that ????? ?+ }.，十(2，+1)11'(λ)2-' ~ ¥ htず斗2"N(A)2-'十 J \...J'b /fJVV./l~ V2 ) ? ???? ??∞ V(λ〉∞三玉.2J--1包-2J V4+ J.ヨ}，=:l 1. '1) =1λ-1 and by the Borel噌Cantellilemma， we hav告 foralmost a11 X v42-Vく∞，{ d δ d I3十l 五百f(川 !41747+y+ }423 Since d>O is an arbitrary number， we obtain (2.4). Proof of Theorem 2 When f(x)昌nd似た}satisfy th母むonditionsof Theorem 2， and if M斗1三月Icくnj<M十N，3 If1f同川川4半R(:Yニ 0(JFL)，then it is seen that .1li十1' ，.1+11'-) ]1+1' ! M ¥ < :E la"I"+ 2 :E a" :E aiR{二五T)k-lfJ十 k-lfJ+l j-l，~ト1\ .L V / j1lZJ山and then (3.1) 手(Z1内2)(1+ ワ;;l~) If we put in the above in日quality5 On tle C01ZZ!ergellu of Some Gat Serie、fllf=sZ呂ndN三 N(sう=2叶 1，fl:丈1MM〉i2drzo(;:ff)then we have and the almost everyw11ere convergence of 同 (8+1)' Lj 2J s開 1k四.，z+1 a"f(1ZkX) (3.2) ???、????? ?????? 、 、? ? ?holds， whenever (3.3) However t11is is reduced from (1，9)， i.日"佃 f($十1)' ¥112 ∞ f C， +1)' ~('$~) a.l，" t4 = ~.L"ij; a，h/k的止 向山一一一一上ープ←)む28渇 i¥k哩i-t+l Wfe 1 咽 l\lc~~ず+1 "~(i; )1 .~ v V- ò'~ ¥.，"""'61)1"'./ -Vk--/o~g 丘…・・(lOj; pkY J /ζ 1 (ti干1)' 一一一一一一一一一一一一一…一一一一dt1V/J 扇子..二百んiJdvh三千+1α，z'Vk1 〔ー∞ 1 白川32 - y Lj一一一 一一.~.-:;jl-Lj， . 2:; ak 2y1 止 !og 止ー…くlogpk)~ I~ .'::'1 S log J くlogps)P f=l k ""，~+l Uic V n. f，.v.s ~ ¥J，v6pn' .F J<0(1)[岳~'/l/k μg k.....{μilpk)s J/2く∞In order to prove Theorem 2 it is only sufficient to prove that 孟f1822%fIJAJrc/(川 2ぬく閃(3匂4)But the left hand member of (3.4) 18 less than by (3.1) ∞ logN(8)N(')" ρ1 s'+(，+Il2" ?51hg N(ふうf]l ，f)o J /c-瓦町，p/(nkx〉12d;r∞ logN(s) I (s +1)' ∞/臼十1) 、=J51μg N(s) 見 N(s)2- Vし5+1GK下.:::gI\ /c -~+1 a/c" )N(s)μg N(s) /∞ (8 +1)' ¥∞ 1 三三OC;~l k-~+l a，2y1 k log k)= O¥;f)αf下/止 lO，gk 1 <∞ (3.2) Thus if we make use of the Menchov's device [6J， we verify (3.4)ヲ andfrom and (3.4) we complete the proof of Th♀orem 2. Proof of Theorem 3 For the proof of Theorem 3 we need a lemma corresponding to Lemma 1. Lemma 2. If f(心 and{1Z，} satisfy the hypotheses of Theorem 3， then it holds that 4 fIZH1 ， N N2 O lAエIt is easy to prove the fol1owing formula (4.1) Proof. 6 N. MATSUYAMA (4.2) I J1fω )fω )dXI=O(Cfob-jA-1伊-)1n fact， ifk>j， then by the Parseval's relation， we have !j1仇りf(njx)dxl守 JEs|=1 2J c.C8n.lnJ I 1ご玉8く∞，nJ18倫， ー4211cstzjfslz)1/¥ベコ)ロ0((fob-11忌ー かα)This is (4.2)， and since Lp-1(のisnon-decreasing as止→∞， we have Jl山 112HN f1 。 Ik-~L 五三副-f(nωIdx=ぷ+1 Lp之王子。 f(nkx)2dx+足 17t1/(nd〉/(n同〉ぬj Lp-μ〉LP-1(j〉jI.~-l 1'Y 1 ¥ 4一一一一γ +O( 2J 一一一一~ 2J T f'" L，íT_~ /'1.. ;，2ct Lp-1(zノし=-'+1 Lp-1(D k.:T+l Lp-1(止)(IOgp-l(止-j))2α/J¥T I N z+N-r 、42え巧z-+Ol忍'1(何人r〉hjZ1LP寸(n斗ィi+川九ア 1¥72 、=一二二一τ+O{ T ， _2， T _ _ I¥T'Z") Lp_1(z)2 '~\ Lp-1(z)2(lOgp-1N戸 jThus we complete the lemma. If we' proceed along the same line as the proof of Theorem 1， we obtain the proof of Theorem 3， but in this case we must put in (4.1) z=β1=21.. 5. Proof of Theorem 4 We suppose that f(x) and {躍k}satisfy the hypotheses of Theorem 4， and if we put νk=乏 τ=[会〕(5.1) ps，k=〔7-必τ(今主計L)τJ.then for each s (s = 1， 2， 3，……， τ)， 孟|α…Iflf(n8叩 )-sぃ (n.+'tkx)ldx孟(孟。ぃt¥孟f|/(匁 8叩 )-sμ.，(n.叩〉刈M4(2163)1/2(Sfl|/ωーら ω J2dxy/2 On the Conver genceザ SomeGat Series 7 手(214)吋完走γy/2=仰)(孟訓告計)山)匂1)(かy/2<∞ It shows the almost everywhere convergence of each series (5.2) EOGHtk{/(n山 ρ)-Sぃ (n.+匂 kX)} (s=1，2， ・…・，τ〉On the other hand we can easily verify the convergence in the L2・sense of two series 2J a，/(nkX) and (5.2)， and then each series is so also. 2J a.+ 't kSμ • ，.(n.+ 't kX) h目。Now by (5.1)， we have 竺ーE土並土工=一色土亘書士L 竺ー主主投士L……一主主士主主土ヱーn.+'tk n.+τk n.+τk+l n.+'tk十勺←1¥、，.- n .+伺'k+i+1¥'"¥、~<..mtn .._.~ .~ j"/Q!1.， k， O!玉i<τ n.+τk+番(s=1，2， ・…・，τ〉and then weobtain by the Kolmogoroff's theorem [7]， the almost everywhere convergence of each series (5.3) ;50 0smSh・.(n.+'tkX)' (s=1，2，...， τ〉whence Theorem 4 follows from (5.2) and (5.3). References [1] M. Kac， R. Salem and A. Zygmund， A gap theorems， Trans. Amer. Math. Soc.， 63(19.崎).[2] S. Izumi， Notes on Fourier Analysis (XLI) ; On the strong law of large numbers and gap series， Tohoku Math. Jour.， 3 (1951). [3] J. L. Koksma， A Diophantine property of some summable functions， Indag， Math.， 13 (1951). Also Sur les suites (;.nx) et les fonctions g(t) E L2， Jour. de Math. Pures et Appliquees， 35 (1956). [4] P. Erdos， On the strong law of large number， Trans. Amer. Math. Soc.， 67(1949). [5] M. Kac， Probability methods in some problems of analysis and number theory， Bull. Amer. Math. Soc.， 55 (1949). [6] S. Kaczmarz and H. Steinhaus， Theorie der Orthogonalreihen p. 162. [7] A. Zygmund， Trigonometrical Series， p.251. 

On the Convergence of Some Gap Series

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On the Convergence of Some Gap Series

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