closeto65jjj.cn

TRANSACTIONSOFTHEAMERICANMATHEMATICALSOCIETYVolume163,January1972CONVERGENCE,UNIQUENESS,ANDSUMMABILITYOFMULTIPLETRIGONOMETRICSERIESBYJ.
MARSHALLASH(1)ANDGRANTV.
WELLAND(2)Abstract.
Inthispaperourprimaryinterestisindevelopingfurtherinsightintoconvergencepropertiesofmultipletrigonometricseries,withemphasisontheproblemofuniquenessoftrigonometricseries.
LetEbeasubsetofpositive(Lebesgue)measureofthekdimensionaltorus.
Theprincipalresultisthattheconvergenceofatrigono-metricseriesonEforcestheboundednessofthepartialsumsalmosteverywhereonEwherethesystemofpartialsumsistheoneassociatedwiththesystemofallrectanglessituatedsymmetricallyabouttheorigininthelatticeplanewithsidesparalleltotheaxes.
IfEhasacountablecomplement,thenthepartialsumsareboundedateverypointofE.
Thisresultimpliesauniquenesstheoremfordoubletrigonometricseries,namely,thatifadoubletrigonometricseriesconvergesunrestrictedlyrectangularlytozeroeverywhere,thenallthecoefficientsarezero.
Althoughuniquenessisstillcon-jecturalfordimensionsgreaterthantwo,weobtainpartialresultsandindicatepossiblelinesofattackforthisproblem.
Wecarryoutanextensivecomparisonofvariousmodesofconvergence(e.
g.
,square,triangular,spherical,etc.
).
Anumberofexamplesofpathologicaldoubletrig-onometricseriesaredisplayed,bothtoaccomplishthiscomparisonandtoindicatethe"bestpossible"natureofsomeoftheresultsonthegrowthofpartialsums.
Weobtainsomecompatibilityrelationshipsforsummabilitymethodsandfinallywepresentaresultinvolvingthe(C,a,0)summabilityofmultipleFourierseries.
Introduction.
Themaininterestofthispaperwillbethetheoryofmultipletrigonometricseries.
MultipleFourierseries(themostimportanttypeofmultipletrigonometricseries)willbediscussedonlyinconnectionwiththetheoryofuniquenessandagaininthelastchapter.
Forthedefinitionsofanyunfamiliartermsusedintheintroductionthereaderisreferredto1.
Oneofthemaindifficultiesinmultipleseriesarisesinconnectionwiththeusualconsistencytheoremsforsummationmethods.
Inordertomaintainthevalidityofthetypicaltheorem"convergenceimpliessummability,"eveninthecaseofPoissonsummationonehastohavetheaddedconditionthatallpartialsumsbebounded.
Ifoneattemptstorestricthimselftoregularmethodsofformingthepartialsums,itiseasytoconstructexampleswherethisconditionfails.
However,byintroducingunrestrictedrectangularpartialsums,convergenceofamultipletrigonometricReceivedbytheeditorsJanuary22,1971.
AMS1970subjectclassifications.
Primary42A92,42A48,42A20,42A24,40B05;Secondary40G10,40A05,40D15.
Keywordsandphrases.
Squaresummable,unrestrictedrectangularlyconvergent,sphericallyAbelsummable,(C,0,X)summable,uniqueness,coefficientgrowth,Riemannsummable.
(1)ResearchpartiallysupportedbyNSFGrantNo.
GP-14986.
(2)ResearchpartiallysupportedbyNSFGrantGP-9123.
CopyrightD1972,AmericanMathematicalSociety401402J.
M.
ASHANDG.
V.
WELLAND[Januaryseriesonasetofpositivemeasureimpliesthepointwiseboundednessofthepartialsumsalmosteverywhereinthegivenset.
Hence,oneachievesawiderangeofconsistencytheoremsonanalmosteverywherebasis.
ThetechniqueofobtainingtheboundednessofthepartialsumshasasitsoriginaworkofP.
J.
Cohen[2],inwhichtheauthorobtainsanestimatefortherateofincreaseofcoefficientsofamultipletrigonometricseriesconvergentalmosteverywherebyaregularmethod.
Byapplyingthistechnique(itisdescribedandappliedin11)inthecaseofunrestrictedrectangularconvergenceinasetofpositivemeasure,oneisabletoevenconcludethatthecoefficientsarebounded(Theorem2.
2).
Lemma2.
1,part(b)ofLemma2.
2,andthefirststatementinTheorem2.
1wereprovedbyP.
J.
Cohenin[2].
Wereproducetheproofsheresince[2]isnoteasilyavailableandsincetheotherapplicationsshowCohen'stechniquetobemorepowerfulandusefulthanhadpreviouslybeenapparent.
Anotherdifficultyinmultipleseriesisthediversityofpossiblepartialsums.
Asispointedoutin111,thisdiversityintroducesproblemsatthemostfundamentallevel.
Perhapsthemost"natural"methodsofformingpartialsumsarebycircles,squares,rectangles,anddiamonds(correspondingtodiamondsistriangularconvergence).
Severalexamplesaregivenin111toshowthebasicincompatibilitiesbetweenthesemethods.
SeeFigure3forasummaryofthesituation.
Thisin-compatibilitymakessomewhatsurprisingthefactthatconvergence(=unrestrictedrectangularconvergence)everywhereimpliessphericalAbelsummabilityevery-where(Theorem3.
1).
InIV,thefactfrom11thateverywhereconvergencecontrolsthegrowthofthecoefficientsandthefactfrom111thatconvergenceimpliessphericalAbelsum-mabilityarecombinedwithauniquenesstheoremofV.
Shapiro[16]toobtainuniquenessfordoubletrigonometricseriesthatconvergeinasetwhichexcludesatmostonepoint.
Thequestionofuniquenessunderthehypothesisofeverywhereconvergenceisstillopenindimensionsgreaterthantwo.
Wegivethreeotherpossibleapproacheswithpartialresults.
Oneresult(Theorem4.
3)isthatunique-nessholdsformultipletrigonometricseriesofpowerseriestypethatconvergeeverywhere.
FinallyinVsomerecentlyprovedresultsinonedimensionalFourierseries[1],[9]areusedtoprove(C,a,0)summabilityforcertaindoubleFourierseries(a>0).
ThisresultcomplementscertainrecentresultsofC.
Fefferman[6],[7],P.
Sjolin[19],andN.
Tevzadze[20]concerningtheconvergenceanddivergenceofmultipleFourierseries.
Forfurtherreferencesthereaderisreferredtotheexcellentbibliographyin[18].
I.
Definitions.
Inthispaper,wewillbeconsideringvariousaspectsofpointwiseconvergenceofamultipletrigonometricseries(1.
1)T(x)=>ameim*x.
1972]CONVERGENCE,UNIQUENESSANDSUMMABILITY403Weusem=(m1,M2.
.
.
,mJ)wherethemiareintegersandx=(x1,x2,.
.
.
,Xk)ETk.
WewillunderstandTk=[0,277)x[0,27T)xk*x[0,27T)tobeEkwithtwovectorsxandyidentifiedwheneverxi-yi(mod2i),i=1,.
.
.
,k.
Also,Ix-yI=min{[I.
(X*-y*)2]12X*iSequivalenttoxandy*isequivalenttoy}.
ForzeC,Izlmeanstheusualmodulusofacomplexnumber.
Wesetmx=im1x1+.
+MkXk,ImJ=(M2+.
.
.
+M2)1)2,lmil=mini{lmil},andllml11=maxi{JImil}.
Therelationm>nwillmeanmi_ni,i=1,.
.
.
,k.
LetQkbethesetofallk-tuplesP=(Pl,,Pk)wherepi=Oor1fori=1,.
.
.
,k.
IfpEQk,wesetf@x)=iaXi-axi.
fand(dx)P=dxi.
dxi2,.
dxiwherepi,=1,s=1,.
.
.
,randpj=0otherwise.
Whenk=2,wewilltakem=(m,n),x=(x,y),Iml=(M2+n2)1"2andsoforthinordertosimplifythenotation.
Therearemanyinterpretationsofthestatement"Tconvergesatx.
"Amethodofconvergenceisdescribedbyasequence{En}n=0,1,2,.
.
.
offinitesetsofk-tuplesofintegerssuchthateachEniscontainedinEn,+forallsufficientlylargejandtheunionofalltheEnconsistsofallk-tuplesofintegers.
ThemethodissymmetricifmEEnimpliesthatsoaretheotherk-tuplesm'satisfyingIm'l=lm,I,i=1,.
.
.
,k.
Inthispaperonlysymmetricmethodswillbeconsidered.
AmethodofconvergenceisregularifandonlyifthereisaconstantKsuchthatforeverylatticepointm=(m1,mk)thereexistsnosuchthatmbelongstoEnoandsuchthatforeachlatticepointIinEnowehaveIIIljII'KIIImIII.
WecalltheleastsuchKtheeccentricityofthemethod.
Inotherwords,theEn'sarenottooeccen-tricallyshaped.
ThisdefinitionisduetoPaulJ.
Cohen[2,p.
39].
Someexamplesofregularmethodsofconvergencearesphericalconver-genceorcircularconvergenceifk=2(En={mIImj0andiflimt,0T(x,t)=s(x).
WesayTisRiemannsummabletos(x)if(sin_m1h1\2sim2h2\2(sinmkhk\T(x;h1,l.
.
,hk)=,ame2xm1h1m2h)mJhkiconvergesabsolutelyforh2++h2=hIhi2=0(interpret(sinO)/Otobe1),andifliMtlhl-oT(x;hi,.
.
.
,5hk)=S(X).
WewillconsideronlyLebesguemeasurablesubsetsAofTkandJAIwilldenotetheLebesguemeasureofA.
II.
Relationshipsbetweenmethodsofconvergenceandthegrowthofcoefficients.
LEMMA2.
1(P.
J.
COHEN[2]).
LetScT1withISI>.
Thenforanyinteger1>0thereareIpointsofS;z1,zl,suchthatizi-zjl>(8/I)foriAfj.
Proof.
Wechoosezp,p=1,2,recursivelyasfollows.
Letz1beanypointofS.
ForthisproofonlyidentifyT1with[z1,z1+27r).
ChooseZ2>Zl+8/1SOthatISn[z1,z2]Izp1+8/isothatISn[zp-1,zp]IISI-(p-2)'SI>2.
ObservethatlSn[z1,z1]l(z+27)-z_ISn(z1,zlI+2r)_I3ISl/1>71sothatz1andz1aresufficientlyseparatedaspointsofTV.
Itisclearfromtheconstructionthatanyotherpairofpointsarealsosufficientlyseparated.
LEMMA2.
2.
Letp(w)=ao+alw+*a,wnbeatrigonometricpolynomialwherew=eixrangesovertheunitcircle.
(a)IfIp(w)I0,thenthereisaconstantc=c(jEj)suchthatIp(w)I1,thereexistsA=A(y)2rA,thenforallwET',(2.
1)jp(w)j1,bepointsof{zeC:IzI=1}suchthatIargzi-argzjI>2vrA/(n+1)ifij.
SincewewillusetheLagrangeinterpolationformulap(z)-:=Ep(zOi)i(z)where(2.
2)vri(z)=rli(Z-z,)weneedtoestimatethe77i(z).
Becauseofsymmetry,itsufficestoestimate7r0(z).
Withoutlossofgenerality,weassumethat0=argzO|(1-g)j=lC,eAfor|1-zll|1-CJ,J1-z,nJ>J11n1-J-J1-z21>Z1-11J,ll-Zn-11>11t-2etc.
Wealsohave(2.
5)Fl(z-zj)6n(4k)=(4/m)26n(8n)!
2>(4/m)26n((2T)112(8n)6n+2e-6n)2ThislastinequalityfollowsfromStirling'sinequality.
Combiningthiswith(2.
6)and(2.
8)wehave(2.
9)17rro(z)J2iTA,byLemma2.
1wemaychoosezo,z1,.
.
.
,ZnfromEwhereIargzi-argzjI>2rA/(n+1)ifiOj.
AssumepisboundedbyBonE,andrecallp(z)=Op(zj)rj(z).
By(2.
9)andthecommentfollowing(2.
2)wehave,forallz,0sothat,by(2.
7),>0isboundedawayfromzero.
Hence,Supn,yy(n,8)=c.
Toprove(b),firstchooseAsocloseto1andthennosolargethaty(n,8)>yifnno.
Setb=maxkl=.
no{y(S,k)k,I}.
From(2.
10)wehaveIp(z)Ip.
andyeFO,(Tn(XYo)fp.
andxEFxo,whereFxocEXOandF,0c-E,havepositivemeasure.
Further,chooseBsolargethat(2.
12)ITmn(XO,yo)I-BifmandnarebothlessthanH.
.
Thiscanbedonesince(2.
12)simplydemandsthatBbebiggerthan(p.
+1)2numbers.
WestillhavetostudyTmn(Xo,Yo)whenm>p>norn>,u>m.
Thetwocasesaresymmetrical;sohenceforth,assumem>p>n.
From(2.
11),ITm(xo,y)j2,thetheoremcanbeprovedbyaninductionwhichrequiresnonewideasnotalreadypresentinthek=2case.
Onesimplyreplacesmby(n1,.
.
.
,nk-),nbynkandmakesasimilardecompositionofF.
ForcompletenessweincludeaproofoftheCantor-LebesgueTheorem.
OnecouldalsoseeRevesandSzasz[14,pp.
693-695]orGeiringer[8,p.
69].
Forn>0,An(X)isthesumofalltheelementsof{a,exp[il.
x]:41=ni,i=1,.
.
.
,k}.
408J.
M.
ASHANDG.
V.
WELLAND[JanuaryCANTOR-LEBESGUETHEOREM.
IflimminnjA,(x)=0forallxEE,IE>0,thenlin1linij,xan=Proof.
Weproceedbyinduction.
ForthetheoreminonedimensionseeZygmund[23,Vol.
I,ChapterIX].
Assumethetheoremholdsfork-IandthatReal(An(x))=an(x')cosnkXk+bn(x')sinnkXk=p(n;x')cos(nkXk+c(n;x'))=o(1)as1nooforeveryx=(x',Xk)EE.
Itsufficestoshowthatp(n;x')-O0foreveryx'insomesetFofpositivemeasure.
LetF={x'f'L,,XE(X,Xk)dXk>0},whereXEisthecharacteristicfunctionofE.
ByFubini'stheorem,IFI>0.
Foranyfixedx'EF,(2.
16)p2cos2(nkXk+a)-0forallXkE{Xk(X',Xk)eE}=F(x').
ButbydefinitionofF,IF(x')I>0,sointegrating(2.
16)overF(x')showsthat4-p2IF(x')I,andhencealsop,tendstoOas-1noo.
THEOREM2.
1.
IfT(x)isconvergentalmosteverywherebyaregularmethodofconvergenceofeccentricityk,andify>1isgiven,thenthereexistsb=b(T,y,k)>0suchthat(2.
17)IamIbylllmlll.
IfT(x)isconvergentonasetEofpositiveLebesguemeasureIE,thenthereisc=c(T,IEl,k)>1suchthat(2.
18)IamIcIllmIll.
Theseresultsareinasensebestpossible.
Proof.
Lety>1begiven.
LetSn(X)=JEiEcjeiixbethenthpartialsumofT(x)withrespecttothegivenregularmethodofsummation.
Wearegiventhat{sn(x)}convergesforalmosteveryxETk;hence,thatthesn(x)areuniformlyboundedonarbitrarylargesubsetsofTk.
Considersn(x)asapolynomialinonecomplexvariableexp(ix1)=z1withcoefficientsdependingonx'=(x2,.
.
.
,Xk),JI~~~~~~~~2JIIsn(x1,x')I=2d(x')exp(ijx1)=Edm-j1(x')exp(imx1)j=-JIm=oWemayassumethatSnisboundedbyaconstantBAonasetoftheformUX'SH(G(x')x{x'})whereIHI>(27rA)k-1;foreveryx'eH,IG(x)I>(27A)andAsupi.
kJi,jEEn-Hence,|c;I_Bx(b(yj))kY2kK111111.
Ifwesetyl=yl12kKandb=BAb(yl)k,thisprovesthefirstpartofTheorem2.
1.
ThesecondpartofTheorem2.
1followsexactlythesamelineofproofexceptthatpart(a)ofLemma2.
2isusedinplaceofpart(b).
P.
J.
Cohen[2,p.
44]givesexamplesoftrigonometricserieswhosecoefficientsare"almost"0(111mjll).
ToshowthatTheorem2.
1isbestpossiblewegivenewexamples.
Consideratwodimensionalseriesoftheform00(2.
19)t(x,y)=20(n)(1-cosx)neinn=1wherethechoiceofbwilldeterminethepropertiesoft.
Since2n2n12n\(2.
20)(1-cosx)n=2sin)=2n>Jei(n)x(_1)n-'theNthpartialsumof(2.
19)maybewrittenNnl2nNN(2.
21)tN(X,y)=221)-I(n)2-'(neimx.
e=Etmnen=1m=-nnMnNmNFrom(2.
21)itisclearthattheordinaryNthpartialsumoftcorrespondstothe(N,N)squarepartialsumoftthoughtofasadoubleseries.
WenotethatIto,nI=k(n)2-n(T)=_)2_(2.
22)nj2an)112(n2ne-2n/2n0n>k(n)2-2nn'~if)2V(5/4)(2-rn)12nne-n)2T0)3n1/23n12Nowset0(n)=3n112.
Foreachxsatisfying1-cosxl2nwhichshowsthat(2.
18)cannotbeimproved.
Toseethatthe"almosteverywhere"hypothesisinthefirstpartofTheorem2.
1cannotbeweakened,set-(n)=3n"12r-n.
Givenanye>0,r(2r)2-E.
Nevertheless,Ito,nl>(2/r)n,sothatthecoefficientsstillgrowexponentially.
Finally,giventhataseriesdoesconvergealmosteverywherewithrespecttosomeregularmethod,thesomewhatawkwardcondition(2.
17)isthemostthatone410J.
M.
ASHANDG.
V.
WELLAND[Januarycandeduceconcerningthegrowthofitscoefficients.
LetN=III(m,n)111=max{jml,Inj}.
Assumethattm,nissquareconvergentalmosteverywhere.
Then(2.
17)assertsthat(2.
23)forally>1,Itm,nIIyN1,limItm,n0.
N-oFor(2.
24)obviouslyimplies(2.
23)andconverselyassuming(2.
23)andgiveny>1,wehave,pickingy'betweenyand1,Itm,nI-t(/\1,Ob(N)IyN>-0,choose-(n)=3n"2'0(n)2-n.
Thenalmosteverywhere(infactexceptforthetwolinesX=7randx=-,r),t(x,y)convergesascanbeseenbysettingy2=2/(l-cosx)ifx.
rr(ifx_0,t(0,y)=0).
ButIto0,nI>+/(n)sothecoefficientsgrowfasterthanb.
Forconcreteexamplesthereadermightconsider0(n)=n1O6or0(n)=2n1(logn.
Inotherwords,almosteverywheresquareconvergencepermitsthecoefficientsanyrateofgrowthwhichislessthanexponential.
REMARK.
Lemma2.
2isalsofairlysharp.
LetP(z)=einx((l-cosx)/2)n.
Pisapolynomialofdegree2n.
IfIP(z)IBon[-8,8],B(8/2)2nandsupIP(z)I=1sothatsupIP(z)jI-B(4/1[-8,8]1)2n.
Thisshowsthatthec=c(lSI)ofLemma2.
2cannotbechosensmallerthan4/1S.
AclosereadingoftheproofofLemma2.
2showsthatcmaybechosenoftheformdlSIbutwedonotknowifdmaybeassmallas4.
THEOREM2.
2.
IfT(x)isconvergentonasetE,IEJ>0,then(2.
25)am=o(1)asI1mliooand(2.
26)am=0(1)forallm.
Thisisalsoabestpossibleresult.
Proof.
GivenanumericalseriesijOai,withpartialsumsSn=&>jiOai,itcanbeseenthat(2.
27)an=-(-1)s6.
Inthisformula(-1)"-(1)61+62++6kandweunderstandSn_atobeequalto0wheneveranynt-5i=-1.
Inoneandtwodimensionsthisreducestotheelementaryfactsthatan=Sn-Sn-,andam,n=Sm,n-Sm-l,n-Sm,n-1+Sm-l,n-i.
ThelastequationmaybevisualizedasA=(A+B+C+D)-(B+D)-(C+D)+DwhereA=amn,B=m-akn,C=o&am,andD==0-akl.
19721CONVERGENCE,UNIQUENESSANDSUMMABILITY411BADCFIGURE1SincethisfiguresomewhatresemblesaMondrianpainting[5,p.
169],wewillrefertoanapplicationof(2.
27)asMondrianing.
LetAj(x)=i=j,cleilxwherej>0.
IfTconvergesatxtos,writingTn(x)Aj(x),n_j>oexpressingAn(x)intermsofpartialsumsasin(2.
27),observingthatifallindicesarelargeeverypartialsumin(2.
27)isclosetosandthat(2.
27)hasanequalnumberofpositivelyandnegativelysignedterms,wefindthat(2.
28)limAn(x)=0.
minni-oDFromthehypothesisofTheorem2.
2,itfollowsthat(2.
28)holdsforeveryxinasetofpositivemeasure.
From(2.
28)andtheCantor-LebesgueTheoremitfollowsthatlim11n11_.
an=O,whichis(2.
25).
Wenowprove(2.
26).
FromLemma2.
3and(2.
27)wehavethatforeachxofF,thereisaconstantC(x)suchthat(2.
29)IAj(x)lC(x)forallj>0.
SinceFhaspositivemeasurewemayfindasubsetF'cFofpositivemeasureonwhich(2.
29)holdsuniformly,thatis,(2.
30)1Aj(x)I0.
FromthisitfollowsbythesameargumentusedintheproofoftheCantor-LebesgueTheoremthatwemustalsohave(2.
31)1CjI2,atthiswriting.
III.
Relationshipsbetweenmodesofconvergenceandsummability.
THEOREM3.
1.
If(3.
1)T(x)=Eei"convergesateachxofasetE,thenT(x)issphericallyAbelsummableatalmosteveryxinE.
Inparticular,ifthecomplementofEisempty(orevencountable),thentheconclusionholdseverywhereonE.
1972]CONVERGENCE,UNIQUENESSANDSUMMABILITY413Proof.
AnapplicationofLemma2.
3showsthatitissufficienttoprovethefollow-inglemmaconcerningnumericalseries.
LEMMA3.
1.
If>m>0cmconvergestosandhasallpartialsumsbounded,thenitissphericallyAbelsummabletos;thatis,Zm>oc_eImIlh=s(h)existsforeveryh>0andlimhOs(h)=s.
Thislemmais,inturn,aconsequenceofthefollowinglemmaofstandardtype.
Beforewestatethelemmawemustdevelopsomeadditionalnotation.
LetpEQkwithl'sinthei1,i2,i,placesandO'sintheremainingk-rplaces.
Then00co00>jam=EE***am(p)mj1=m12=mtr=Owheremjisfixedifj{i1,i2,.
il,}.
Inparticular,ifp=0,:(p)am=am.
Further,000000Ztam=ZL.
.
.
Z**Ljam(p)mt,=Omisj1=Omis+1`=mir=Qwheretheprimeindicatesthatoneofthesummationshasbeenomitted.
Ifp=0,set2(p)am=0.
Notethatifp#0andifallthemjwithj0{i,i}arefixed,2(p)amdenotesasinglesumwhilepamdenotesaninfinitefamilyofsums,oneforeachchoiceofsandmin,s=1,2,.
.
.
,r,m,=0,1,2,.
Ifp#0,wedefinel\PAmtobeAiA.
.
AiAwhereAiAA=-AM"MilMiM+1.
.
Mk-AMJ1,M+1^+1MkandA0Am=Am.
LEMMA3.
2.
LetY2m>Ocmconvergetosandhaveallpartialsumsbounded.
Then,cmAm(h)existsforallh>0andlimh-c_cmAm(h)=sprovidedthefunctionsAm(h)satisfy(3.
2a)JLPAm(h)l0)andofthevaluesofthemj'scorrespondingtotheindicesnotsummed;(3.
2b)limIIPAm(h)I=0h-O(p)foreachfixedpEcQk,fixedchoiceofomittedsummation,andfixedsetofmj'scorrespondingtotheindicesnotsummed;and(3.
2c)Ao(h)=1.
Proof.
TheproofisadirectapplicationofAbel'spartialsummationformula:kNj+1NilNi2N,t(3.
3)>:anAn=ZZ5.
2ml\PAm.
i=01ni=OPCfknt,=On2=nir=OHereSm=Xm>u>oauwheremi1=ni1,.
.
.
,min=ni,,m}=N+1ifj{i1,.
.
.
,ij}Weapply(3.
3)witham=cm,Am=Am(h).
Wearegiven(3.
4)5sm1B.
414J.
M.
ASHANDG.
V.
WELLAND[JanuaryGivenanye>0thereisanumber,usuchthat(3.
5)iSm-S1,twherecistheconstantof(3.
2a).
In(3.
3)ifweputao=1,am=0form#0,weobtainN1Nr(3.
6)1=Ao(h)=.
_APAm(h).
(p)nil=Onr=OThisfollowssince(3.
2c)holdsandSm=1foreverym.
Byvirtueof(3.
6)wemayassumes=0andreplace(3.
5)by(3.
7)ISmI,.
NowN1Nrn,l=nir01-1-0000-0000-Combining(3.
9)ifpA0and(3.
10)ifp=0withAbel'spartialsummationformula,weseethatifE>0isgiven,onemaychoosefirstU=,U(E)sufficientlylargeandthenho=ho((,e)=ho(e)sufficientlysmallsothat,wheneverNl,.
.
.
,Nk>and00,choosehsosmallthat(n,+1+l)hame.
i(mx+E=2m,n=-a:WheneverIsinyI11/2whichdoesnottendtozero.
AnotherinterestingexampleduetoC.
Fefferman[6]assertsthatthereexistsacontinuousfunctionoftwovariableswhichhasaFourierserieswhichdoesnotconvergealmosteverywhereforrestrictedrectangularsums.
Thisexamplealso1972]CONVERGENCE,UNIQUENESSANDSUMMABILITY417separatessquareandrestrictedrectangularconvergence,sinceFefferman[7],P.
Sjolin[19],andN.
Tevzadze[20]haveshownthattheFourierseriesofafunctioninL2convergesalmosteverywhereforsquaresums.
Oneshouldnotethata(x,y)isnotaFourierseriessinceby(3.
22)theam,,,donottendtozero.
EXAMPLE3.
2.
Thereisaserieswhichisrestrictedlyrectangularlyconvergentatalmosteverypointbutwhichistriangularlydivergenteverywhere.
ConsidertheseriesO000(3.
23)b(x,y)=>21exp(i4'x)sin21y=zbm,nei(mx+nY).
l=lm,n=-1Foreachfixed1,theseriescontributestermswithindices(3.
24)(41,-21'),(41,-21+2),(41,-21+4)41,21).
Inthelatticeplane(seeFigure2)allofthesepointsarecontainedwithinthesetH={(m,n)IImln2}.
Letaneccentricitye>1bespecified.
Sinceif1issufficientlylargeallofthepointsof(3.
24)lieoutsideoftheregionR={(m,n)Ie-1(41)112),itfollowsthattherestrictedlyrectangularpartialsumsofeccentricityboundedbyeeventuallycoincidewiththeordinarypartialsumsofb(x,y).
These,inturn,convergealmosteverywhereforifIsinyIwhichcontradicts(3.
26).
EXAMPLE3.
3.
Thereisaserieswhichisrestrictedlyrectangularlyconvergentatalmosteverypointbutwhichiscircularlydivergenteverywhere.
Thesameexampleb(x,y)mentionedabove(see(3.
23))alsoiscircularlydiver-gent.
TheproofissimilartothatoftriangulardivergencegiveninExample3.
2.
Fix(xo,yo)andlet(3.
28)S(R)=Ebm,nexp(i(mx0+nyo)).
lml+)n]R418J.
M.
ASHANDG.
V.
WELLAND[Januaryn=emn/FIGURE2Asabove,toseethatlimR,O,S(R)doesnotexistitsufficestoshowthat(3.
29)S(421)-S(42-1)=b4'0exp(i4'x).
Here,wewishtopointoutthatExample2.
1isevenstrongerthanExample3.
3sincethetheoremsofR.
Cooke[3]andA.
Zygmund[25]show(2.
33)canconvergecircularlyonlyinasetofzeromeasuresinceitscoefficientsdonottendtozero.
Example2.
1isastrongerexamplesinceitnotonlyconvergesrestrictedlybutalsoconvergesunrestrictedlyalmosteverywhere.
TheadvantageofExample3.
3isthatitiseasytoseethatitiscircularlydivergenteverywherewhilethesetofcirculardivergenceofExample2.
1isnotknowntous.
EXAMPLE3.
4.
InExample3.
2,onesawthatthesquarepartialsumsofb(x,y)convergealmosteverywhereandthatthecoefficientsofb(x,y)donottendtozero.
Byrotatingthetoruswiththechangeofvariablesx=x'+y'andy=x'-y'weobtainaseriesc(x',y')whichistriangularlyconvergentalmosteverywhere.
However,sincethecoefficientsdonottendtozero,c(x',y')doesnotconvergecircularlyinasetofpositivemeasurebytheCooke-Zygmundresult.
Furthermore,thereasoningusedinExample3.
2showsthatc(x',y')isnowhereconvergentforsquarepartialsums.
1972]CONVERGENCE,UNIQUENESSANDSUMMABILITY419EXAMPLE3.
5.
Thereisaserieswhichisconvergentatalmosteverypointbutwhichistriangularlydivergentalmosteverywhere.
ToseethisweshalldefineaseriesL(x,y)=M(x)8(y)ofthetypeconsideredinExample2.
1.
Asbefore,8(y)=1+2Zn=1cosny.
WechooseM(x)=nbnsinnxtobetheFourier-StieltjesseriesofaLebesguefunctionF(x).
ThefunctionF(x)isdefinedbyF(x)=limp.
OFp(x)for-TTxTwherethecontinuousfunctionFp(x)satisfiesFp(O)=0,Fp(7r)=Tr/2;Fpincreaseslinearlyby2-P(7r/2)oneachofthe2Pintervalsoflength(2/5)Pwithleft-handendpointaco+al(2)++ap(25)P-lwhereeacha:is0or(-)-r;Fp(x)isconstantelsewhereon[0,-r];andFpisextendedto[-Tr,0]byFp(x)=Fp(-x).
Acalculationsimilartotheonein[23,Vol.
I,p.
195]showsthatbn=Oifnisevenand00bn=1)(n-1)/2j7cos(-1--Tn(2)k)k=OifnisoddsothatinanycaseIbnl_Iy(--5n)Iwherey(u)=i=Ocos(7UC(2)k),sothatbnO--0[23,Vol.
II,pp.
147-148,andTheorem11.
16onp.
151].
ItfollowsthatM(x)convergestozeroalmosteverywherebutnoteverywhere[23,Vol.
I,pp.
347-348,especiallythesufficiencypartofTheorem6.
8andthestatementofTheorem6.
11].
TheconjugateseriesM(x)=-1bncosnxconvergesalmosteverywhere[23,Vol.
II,p.
216,Theorem4.
1]andforeveryy,I{xIM(x)=y}I=0.
Thisissoforotherwisethefunctionu(z)=u(retx)=En=1bn(reix)nwhichisanalyticonlzl0)wouldhaveaconstantnontangentiallimitonasubsetofIzl=1ofpositivelinearmeasure[23,Vol.
I,p.
100,Theorem7.
6;p.
105,8,firsttwosentences;andp.
253,Theorem1.
6]whichwouldforceallbntobezero[23,Vol.
II,p.
203,Theorem1.
9]whichisimpossiblesinceM(x)doesnotconvergetozeroeverywhere.
BytheargumentusedinExample2.
1itfollowsthatL(x,y)isconvergentalmosteverywhere.
ToshowthatL(x,y)istriangularlydivergentalmosteverywhere,itsufficestoshowthatAN(X,y)=tN(X,y)-tN-l(X,y)=m,nei(mx+ny)Iml+Jnl=Ndoesnottendtozeroatalmostevery(x,y)ET2.
UsingseveralelementarytrigonometricidentitieswefindthatN-1AN(X,y)=>bmsinmx(2cos(N-m)y)+bNsinNxm=1N-1=2Ebmsinmx(cosNycosmy+sinNysinmy)+o(1)m-1N-1=cosNyEbm{sinm(x-y)+sinm(x+y)}m=1N-1+sinNy>bm{cosm(x-y)-cosm(x+y)}+o(1).
m=1420J.
M.
ASHANDG.
V.
WELLAND[JanuaryForalmostevery(x,y)ET2,M(x-y)=M(x+y)=OandM(x-y)andM(x+y)exist.
Forsuchan(x,y),wehaveAN(x,y)=o(l)+sinNy{M(xy)-M(x-y)+o(l)}+o(l)=sinNy{M(x+y)-M(x-y)}+o(l).
SinceM~isnotconstantonanysetofpositivemeasure,thequantityinbracesisnotzeroalmosteverywhere.
Sinceforalmosteveryy,sinNydoesnottendtozero[23,Vol.
I,p.
142,CorollarytoTheorem4.
27],neitherdoesA,(X,y)aswastobeshown.
ThisexampleisstrongerthanExample3.
2,butissomewhatlessconstructivesinceherethesetsofconvergenceanddivergencearenotspecified.
REMARKSANDPROBLEMS.
Itshouldbementionedexplicitlythattheaboveexamplesshowmostcommonregularmethodsarepairwiseinequivalent,evenonanalmosteverywherebasis.
Forexample,b(x,y)isalmosteverywheresquareconvergentbutisnowherecircularlyconvergent.
Wedonotknowofexamplesofserieswhicharecircularlyconvergentbutnotsquare(or,equivalently,triangularly)convergentonasetofpositivemeasure.
ThesituationmaybesummarizedbythefollowingdiagraminwhichA-F1BmeansthatthereisatrigonometricseriesconvergentalmosteverywherewithrespecttomethodAbutconvergentonnosetofpositivemeasurewithrespecttomethodB,whileC->DmeansthatconvergencewithrespecttomethodConasetforcesconvergencewithrespecttomethodDalmosteverywhereonthatset.
CONVERGENCE(UnrestrictedRectangulIar)_RESTRICTEDTRIANGULAR>SPHERICALRECTANGULAR\/ISQUARE;'FIGURE31972]CONVERGENCE,UNIQUENESSANDSUMMABILITY421Despitethebasicincompatibilitybetweenvariousmethodsofconvergence,ifonestickstoagivenmethod,consistencytheoremsoftheform"convergenceimpliessummability"areoftentrue.
Forexample,ifthenumericalseries0000(3.
30)aiji=oj=oissquareconvergenttos,thenthe(C,1)means(3.
31),1whereSm,n,=:i=o=%aij,alsoconvergetos.
Similartheoremsholdforsphericalandtriangularpartialsumsforherealsothesituationisessentiallyone-dimensional.
However,onesometimesneedstheadditionalassumptionofboundednessofthepartialsumsforconsistency.
Forexample,thenumericalseries2bi,whereboj=1,j=1,2,bij=-1,i=1,2,bij=0otherwiseissquareconvergenttoOsinceallsn,=0buthas(C,1,1)meansI1nn(n+2)(3.
32)Ufln,fn(n+1)2i=0-6(n+1)andsoisnot(C,1,1)squaresummable.
ThisisthemotivationbehindsayingthataseriesconvergesinthesenseofPringsheimifitisunrestrictedlyrectangularlyconvergentandhasboundedpartialsums.
AtypicalconsistencytheoremisLEMMA3.
3.
Ifthenumericalseries(3.
30)convergesinthesenseofPringsheimtos,thenitissummable(C,1,1)tos,thatis,(3.
33)limUm,n-lim1mn~S=Smin{m,n)-(m+l)(n+1)min{m,n}-a>(m+l)(n+l)i=Qj=OProof.
Givene>0pickMusolargethatIsj-Ssp.
.
ThenpicknandmsolargethattzB/(m+1)ameimx-Imlhh-0andf*(x),theupperAbelsum,isanalogouslydefined.
Thentheseries(4.
1)istheFourierseriesoff*(x).
ApartialuniquenesstheoremfortriangularpartialsumshasbeenobtainedbyGeorgeCross[4].
M.
H.
Nasibov[12]hasproveduniquenesstheoremsforunrestrictedandrestrictedconvergence,buthishypothesesareratherstrong.
Allofthesetheoremsassumesomethingabouttherateofgrowthofthecoefficients.
Thefollowingtheoremavoidsthisproblem,butwehavebeenabletoproveitonlyintwodimensions.
THEOREM4.
2.
Letthetrigonometricseries(4.
3)2am,nei(mx+ny)beconvergenteverywhereonT2tothefinite-valuedLebesgueintegrablefunctionf(x,y).
Then(4.
3)istheFourierseriesoff.
Recallthatconvergentmeansunrestrictedlyrectangularlyconvergent.
Letfo(x,y)bedefinedbyfo(x,Y)=liminf2am,nemin{jmj,jnl}-candf(x,y)bedefinedanalogously.
Letq=(ql,q2)beanypointofT2.
ThenasomewhatmoregeneralversionofTheorem4.
2is1972]CONVERGENCE,UNIQUENESSANDSUMMABILITY423THEOREM4.
2'.
Supposethat(4.
4a)f0(x,y)andfo(x,y)arefinitefor(x,y)inT2-{q},(4.
4b)foisinL1onT2(4.
4c)foisinL1onT2.
Thentheseries(4.
3)istheFourierseriesoffo.
COROLLARY4.
1.
Supposethatthetrigonometricseries(4.
3)convergestozeroonT2.
Thentheseriesisidenticallyzero,thatis,allthecoefcientsarezero.
ThecorollaryisobviouslyTheorem4.
2inthespecialcasewhenf(x,y)=0.
WeproceednowtotheproofofTheorem4.
2.
TheproofisanapplicationofShapiro'stheorem.
FromLemma2.
3andLemma3.
1itisclearthatconditions(4.
2b,c,andd)aresatisfied.
Toseethatcondition(4.
2a)holds,byvirtueofTheorem2.
2,weneedonlyproveLEMMA4.
1.
If(4.
5)Iam,nItuAIm,W11Oandaneinx'IlAP(Wm)l=0z-zOnontang(p)foreachfixedpEQk,fixedchoiceofomittedsummation,andfixedsetofmj'scorrespondingtotheindicesnotsummed;and(4.
13)w=1,wherew=(z1(zO)-1,Z)1).
RefertoLemmas3.
1and3.
2forthenotation.
Thesufficiencyofconditions(4.
1l)-(4.
13)issosimilartotheproofofLemma3.
2thatweomitit.
Condition(4.
13)holdsbydefinition.
ToeasenotationletQ=(1,.
.
.
,1,0,.
.
.
,0)haveitsfirstrentriesequalto1anditsremainingk-rentriesequalto0.
Wemustshow00Go(4.
14).
Q>j&\(w)(n,lnrNr+i.
.
Nk)jOisabsolutelyconvergentin{z:Izll0.
ThesetwofactsareexactlythehypothesesofatheoremofCalderonwhoseconclusionisthatan=Oforeveryn[23,Vol.
II,p.
321,Theorem4.
24].
REMARKS.
Itisanopenquestionwhetherornotauniquenesstheoremholdsforgeneralseriesindimensionsgreaterthantwo.
IndimensionthreeShapiro'stheoremcannotbeusedbecausetheanalogueofLemma4.
1doesnotgothrough.
Infact,onemayeasilyfindnumbersalmnsuchthata,,-n--0as11(1,m,n)lloobutE|almnl|0(R),R-1Iml>un=-itIml>Inj>-A+B+C+D.
Ifiislarge,Sm,nissmallformandn>,tsoDissmallsinceco00JxlsinU21sinV21(4.
23)ImnhkI$fj(sn)sv)dudv2,(4.
42)limZanet'X(mhsinm)Sh+.
nhn=h-~O+m,__=((mh)2nnh(Interpret(Osin0+0sin0)/02tobe1.
)Proof.
WeapplyLemma3.
2to(4.
43)Am,n(h)=(mhsinmh+(nh)innh)fP(mh,nh)wheref(u,v)=(usinu+vsinv)/(u2+v2).
Makinguseofinequalitiesofthetype((4+1)hf(4.
44)JAm,n(h)-Am+l,n(h)l-|mfP(u,v)du,430J.
M.
ASHANDG.
V.
WELLAND[Januaryconditions(3.
2a)arereducedto(4.
45)Jov)f(u,V)dudv-0sothattheintegralisfinite.
ToestimateB,use(4.
51)and(4.
52)toshowthataff/buu=O(1/r),af/lv=O(1/r)anda2f/luav=O(1/r2)atinfinity.
Sincef=O(1/r)atinfinityandtheJacobianisr,itfollowsthattheintegrandis0(1l/rl+(P-2))atinfinityandhencethatBisfinitesincep>2.
Wenowconsider(4.
46).
Usingaf"/lu=pfP-laf/luand(4.
53)wehave(454)JfPldupafdu+pfPlfdu=p(A(v)+B(v)).
1972]CONVERGENCE,UNIQUENESSANDSUMMABILITY431Againmakingthirdorderestimatesforsineandcosinewededucefrom(4.
51)that(4.
55)1.
Sincer=(u2+v2)12>u1andrv,itfollowsfrom(4.
43)and(4.
51)that1,vI2implies(4.
58)B(v)0given,wecanchoosehsmallenoughsothattheintegralin(4.
59)islessthanorequalto2Ia2dudv.
Usingasimpleargumentinvolvingthedefinitionofimproperintegralsandthefinitenessof(4.
45),weseethat(4.
59)tendstozerowitheandhenceh.
Theproofof(4.
60)issymmetricalwiththatof(4.
59).
Theverificationsof(4.
61)and(4.
62)areroutine.
Finally,condition(3.
2c)holdsbytheparentheticalremarkfollowingthestatementofTheorem4.
5.
ThisshowsthatthehypothesesofLemma3.
2aresatisfiedandcompletestheproofofTheorem4.
5.
Observethatp>2wasnecessaryfortheproofof(4.
45)sothatanapproachtothetheoryofuniquenessviaTheorem4.
5wouldprobablyrequireatleastthreeintegrations.
Actually,toavoidcomplicationsarisingfromtermswithnegativeindicesitmightbebettertouseanevennumberof(henceatleastfour)integrations.
432J.
M.
ASHANDG.
V.
WELLAND[JanuaryV.
OnthealmosteverywheresummabilityofdoubleFourierseries.
LetKna(t)betheathFejermeansoftheseries2+cost+cos2t+*.
Wehave1=sin(n4I)tDn(t)=K(t)=2+cost+***+cosnt2sin(t/2)and,ingeneral,Kn(t)=EA~n_Dv(t)/AxwhereA6=(m/)=(1+)(13)!
(/+m)ConsidertheFourierseriesS[f]ofareal-valuedfunctionf(x,y)whichisLebesgueintegrableoverT2.
WesaythatS[f]is(C,a,1)summabletofat(x,y)iflima%'8(f)(x,y)min{m,n}-X(5.
1)1r-lrn2iJ__f(x-s,y-t)Km(s)Kn1(t)dsdt=f(x,y).
min{m,n)-*cov_a_Inparticular,S[f]is(C,ac,0)summabletofat(x,y)iflimora)x,ymin{m,n}-oo(5.
2)1iXX-lrnim2f(x-s,y-t)Kma(s)Dn(t)dsdt=f(x,y).
min{m,n}-.
oo7JJ_In[24],A.
ZygmundshowedthatiffELP(T2),p>1,andifa>0,3>0,thenS[f]is(C,a,1)summabletof(x,y)almosteverywhere.
In[11],B.
Jessen,J.
Marcinkiewicz,andA.
ZygmundgeneralizedtheresultbyweakeningthehypothesisthatfeLP(T2)tofE(Llog+L)(T2),whilestilldrawingthesameconclusion.
ArecentexampleofC.
Fefferman[6]showsthattheresultsofR.
A.
Hunt[9]andL.
Carleson[1]concerningconvergenceofFourierseriesoffunctionsofonevariabledonotextendtounrestrictedoreventorestrictedrectangularconvergenceformultipleFourierseries.
(Fefferman[7],P.
Sjolin[19],andN.
Tevzadze[20]independentlyalsoprovedthatfunctionsinLP(Tk)(p>1)haveconvergentFourierseriesforsquarepartialsums.
)Inotherwords,thereexistsacontinuousfeL2(T2)forwhichS[f]isnotsummable(C,0,0)tofalmosteverywhere.
However,wecangivethefollowingtheorem.
THEOREM5.
1.
If,foralmosteveryxinT',f(x,y)eL(log+L)2(T1(y))andifg(x)=JfT1If(x,y)I(log+If(x,y)1)2dyeL(T1(x)),then,foreverypositivea,fissummable(C,a,0)almosteverywhere.
Inparticular,iffELP(T2),p>1,thecon-clusionholds.
THEOREM5.
2.
Leta>0.
UnderthehypothesesofthefirstpartofTheorem5.
1wehave(5.
3)j_fJa'0(x,y)Jdxdy1,then(5.
4)orau(x,y)IIdxdyoforalmosteveryx[23,Vol.
I,p.
94].
From(5.
6)and(5.
9)wededucethatforheL'(Tl)(5.
10)or*(h)(x)=limsup!
_IKx(t)IIh(x-t)IdtCxIh(x)foralmosteveryx.
LetM2f(x,y)=supj!
fDn(t)f(x,y-t)dt|Foralmosteveryx,f(x,y)eL(log+L)2asafunctionofy.
Forsuchx'swehave(5.
11)M2f(x,y)dy0isaconstantobtaining(5.
14)af'O(f)(x,y)dxdy0begiven.
ChooseAsolargethat(5.
15)AxZ/Ae/2}1-/2}1e/2ora*(f")>e/2,thenE(e)hasmeasure0.
WriteF=UFn-=U(xY)limsupIam,,n(f)-fI>n=1n=1min(m,n}x-:n1972]CONVERGENCE,UNIQUENESSANDSUMMABILITY435SomeFnwouldhavepositivemeasure,sayIF,.
I>3>0.
Pickinge8>>IE(e)J.
Hence,U,rndoesconvergealmosteverywhere.
ThisprovesTheorem5.
1.
ThespaceL(log+L)2isnotthelargestpossiblespacethatmightbeusedinthestatementsofTheorems5.
1and5.
2.
Sjolin[19]hasprovedthatthehypothesisfortheconvergenceoftheFourierseriesinonevariablemaybeweakenedfrom"f(x)belongstoL(log+L)2onT1"to"f(x)belongstoL(log+L)(log+log+L)onT1'".
Itisclearthatasfurtherresultsareobtainedinthetheoryoffunctionsofonevariable,onemaycorrespondinglyweakenthehypothesesinthefirstpartsofTheorems5.
1and5.
2whilestillobtainingthesameconclusions.
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Sjolin,OntheconvergencealmosteverywhereofcertainsingularintegralsandmultipleFourierseries,Ark.
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DEPAULUNIVERSITY,CHICAGO,ILLINOIS60614Currentaddress(Welland):UniversityofMissouri-St.
Louis,St.
Louis,Missouri63121