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Uniqueness of Representation by Trigonometric Series 875 Uniquenessof Representationby Trigonometric Series J.MARSHALL ASH', DePaul University,Chicago, IL 60614 J. MARSHALL ASH receivedhis S.B., S.M., and Ph.D. degreesat theUniver- I"k sityof Chicago where he was a studentof Antoni Zygmund. He wasa Joseph Fels Rittinstructor at Columbia University from 1966 to 1969.Since then he has beenon theDePaul Universityfaculty, where he becamea fullprofessor in 1974and chairedfrom 1985 to 1987.In 1977he wasvisiting professor at StanfordUniversity. His researchinterests have centeredon multiple trigonometricseries, singular integrals, and real analysis,with forays into functionaland numericalanalysis. Abstract.In 1870Georg Cantor proved that a 2srperiodic complex valued function of a realvariable coincideswith the values of at mostone trigonometric series. We presenthis proof and thensurvey some of themany one dimensionalgeneralizations and extensionsof Cantor'stheorem. We also surveythe situationin higherdimensions, where a greatdeal lessis known. 1. Cantor'suniqueness theorem. In 1870Cantor proved THEOREMC (Cantor [5]). If, forevery real numberx N lim E ce inx = 0, N-?oo n=-N thenall thecomplex numbers cn, n = 0, 1, - 1, 2, - 2,... are zero. Thisis calleda uniquenesstheorem because it has as an immediatecorollary the factthat a 2iTperiodic complex valued function of a realvariable coincides with the values of at most one trigonometricseries. (Proof: Suppose Yaneinx= Ybeeinxfor all x. Form the differenceseries E(an - bn)einxand apply Cantor's theorem.) This theoremis remarkableon twocounts. Cantor's formulation of theproblem in sucha clear,decisive manner was a majormathematical event, given the point of viewprevailing among his contemporaries.2 Equally enjoyable to beholdis therapid resolutionthat we willnow sketch. Cantor'stheorem is relativelyeasy to prove,if, as Cantordid, you have studied Riemann'sbrilliant idea of associating to a generaltrigonometric series T := ECneinx, the formalsecond integral, namely, F(x) = En,O(cn/(in)2)einX+ cO(x2/2). For some interestingremarks on the importanceof thisidea, see thevery enjoyable surveyarticle of Zygmund[27]. Definethe secondSchwarz derivative D of a 'The researchpresented here was supported in partby a grantfrom the University Research Council of DePaul University. The authoris gratefulto one refereefor proposing an expandedtreatment of multipletrigonometric seriesand to theother referee for making careful corrections and adding some historical remarks. 2In theeighteenth century, physicists just "did" Fourierseries (often quite successfully) without worryingabout convergence very much at all. Whendoubts about convergence began to arisein the nineteenthcentury, the first attempts at rigorwere rather heavy handed. See Dauben[9, pp. 6-31] foran interestingaiscussion of thehistorical context. 873 874 J. MARSHALL ASH [December functionG(x) by - DG(x) = rn G(x + h) 2G(x) + G(x - h) h-0 h The stepsof theproof are: 1. Since T convergeseverywhere, it is immediatethat for every value of x, cneifX ? c_nei-nx 0 as n -* oo. By theCantor-Lebesgue theorem, ICnl + Ic_nl O as n -x oo. Appendix1 givesCantor's weak but easy versionof this.(For a statementof themore powerful Cantor-Lebesgue theorem see thesurvey article by RogerCooke [8]. The proof given there is muchshorter than the one in Appendix1, but requiressome of themachinery of modem analysis.) 2. By theWeierstrass M-test from |c__inx icnx) SUp(|C | + I_e 2 + )2 nI (in in)2 n2 it followsthat F(x) - co(x2/2)is a continuousfunction and thatF is theuniform limitof itspartial sums. (See Theorems25.7 and 24.3in [14]for the M-test.) 3. An importantresult of Schwarz's states that if G is continuousand DG(x) = 0 forall x, thenG is a linearfunction. (See [5,pp. 82-83].)Before presenting a proof of this,Cantor remarks that Schwarz mailed this result to himfrom Zurich.) We givea proofin Appendix2. 4. Since - ei(x+h) - 2eix + ei(x-h) sin h2 =-eix hI 2 (Checkthis.), we have nh 2 F(x + h)-2F(x) + F(x - h) sinm n h2 n*0c0 nh 2 From a0 ? S2?=1an= 0 it followsthat limk OaO + n=1an(sinnk/nk)2 = 0. See Appendix3 forRiemann's summation by parts proof of this. Hence DF(x) = 0 so F(x) = ax + /3for some a and /3. 5. The rightside of theequation co-2 + ax + n3 )2 is bounded.(From 2. aboveit is continuous,hence bounded on [0,2 ], and hence bounded everywhereby periodicity.)Letting x -x oo twice firstshows co = 0 and thena = 0. 6. Fromthe observation made in step2 above,we see thatthe sequence SN(X) :-A + E ei O< InI AN (in) 1989] UNIQUENESS OF REPRESENTATION BY TRIGONOMETRIC SERIES 875 convergesuniformly to 0. Butfor each n # 0, (in)2 2Nx)d = -| SN X) e ixdx Cn 2sr ( forall N > n bythe orthonormality of { einx/ xv }, so lettingN - oo givescn = 0 forall n * 0. (The uniformityof convergence allows the interchange of limitand integral.) Q.E.D. Cantor'sbeautiful theorem suggests a varietyof extensionsand generalizations. The remainingfour sections of thispaper will consider some of these. 2. Summabilityand uniqueness. Can we improveTheorem C by weakeningthe assumptionof convergenceto zeroto an assumptionof beingmerely summable to zero?As is oftenthe case in mathematics,the starting point is a counterexample whichdestroys the "obvious extension." The trigonometricseries Ec einx is said to be Abel summableto s if foreach r, O< r < 1, f(x, r) 2=Ec einxrInI convergesand iflimr f-f(x,r)= s. Let z reix = r(cos x + i sin x). Differentiatethe identity X0 00X A 1 1-z 1-rcosx (cos 9te / , z ) S E=O nx)r\= 1 -z 1 - z ) 1 -2rcos x + r withrespect to x to obtain ? (1-r2)r sin x E (n sin nx)r n 2_2 n=1 (1-2rcosx+r 2)2 If x * 0, as r -> 1-, theright side tends to 0; whileif x = 0, everyterm of f (O, r) is 0, so that f(O, r) = 0, whencelimr - f(O, r) = 0. Thus E?2=Insin nx is everywhere Abel summableto 0. Althoughthis example is unpleasant,it turnsout to be just aboutthe worst thing that can happen. THEOREM V (VERBLUNSKY [25,VOL. I, PP. 352, 383],[22], [23]). If cn/InI-> 0 as InI - oo and Ec einx is Abel summableto 0 at everyx, thenall c, are 0. 3. Higherdimensions. When we move fromone to severaldimensions, the picturebecomes much more cloudy. Here is the land of opportunity.Almost nothingis known;almost every question that the novice might ask turnsout to be an open question.The firstgoal is to mimicCantor's Theorem C. Even this apparentlymodest goal remainsto a largeextent unachieved. The hypothesisthat EC einx convergesto 0 has severaldifferent interpretations in higher dimensions. Mostof thesecan be illustratedin twodimensions, so to ease notationI willrestrict myselfto that case. The basic object will be the double trigonometricseries T be T(x, y) = E(m,n)EZxZcmnei(mx?nY) We definea rectangularpartial sum of to Tmn(XI y) = E c,,ei(Pix+vY)I JL,tt<m, Ivl <n a diamondshaped partial sum to be Tn (XI y) := E cE,,^ei(x+y) I,ul+ Ivl <n 876 J. MARSHALLASH [December and a circularpartial sum to be T(X, Y) = E cUe(F+) L2 + v2 <r2 We freezex and y and makefive different definitions of convergence. If Tr- s as r oo say T is circularlyconvergent to s. If Tn s as r xo say T is triangularlyconvergent to s. If Tn - s as r oo say T is square convergentto s. If Tm - s as min{m, n } oo say T is unrestrictedlyrectangularly convergent to s. If Tmnn s as min{m, n} - oo in such a way that m/n and n/m stayless than e, and if thishappens for each (arbitrarily large) e > 1, say T is restrictedlyrectangu- larlyconvergent to s. To each of thesefive notions of convergencethere corresponds a putative extensionof TheoremC. The firstof theseto be provedwas thefollowing. THEOREMSC (V. Shapiro and R. Cooke [19], [7]). If T(x, y) is circularly convergentto 0 everywhere,then all cmn are 0. In 1957 VictorShapiro proved a two-dimensionalversion of TheoremV which implieda weakversion of Theorem SC thatrequired the additional hypothesis that 1 r E -IIlcmnIl O as rx-o. (1) (r- 1)2< m2 +n2< r2 That thishypothesis was notneeded was a consequenceof a generalizationof the Cantor-Lebesguetheorem due to RogerCooke in 1971.(See TheoremAl below.A surveyof Cooke's theorem and extensions of it by Zygmund [26] and Connes[6] can be foundin the MONTHLYarticle by Cooke [8].) The proofof TheoremSC is modeledafter Verblunsky's Theorem V mentioned above. Boththese theorems carry out theRiemann-Cantor program of integrating twiceand thendifferentiating twice. To getat theideas behind Theorem SC, assume thatT is circularlyconvergent to zeroeverywhere. Write M = (m, n), X = (x, y), M- X = mx + ny, and add the assumption that co = 0. (This simplifiesthe notation,but notthe proof.) Let F(X, t)cmei. lt MO IM1 and let F( X) = limF( X, t). Then formally F(X) = O I{2ei = S and formallyF( X) is a secondintegral of T( X). FromCooke's two-dimensionalCantor-Lebesgue theorem it followsthat equa- tion(1) holds.This guarantees the existence of F(X, t). That F(X) existsfor all X followsfrom the convergence and consequentcircular Abel summabilityof T [18, 1989] UNIQUENESS OF REPRESENTATION BY TRIGONOMETRIC SERIES 877 pp. 67-68]. We have ICM12 0ICI? 0 k CI M#O MI IMI=1 k=2 (k - 1k-< MI <k Condition (1) implies ICMI= o(IMI) so applyingfirst this and then(1) itselfshows the curly bracketed sum to be o(k2). Hence the sum is finite,so that by the Riesz-FischerTheorem [20, p. 248] S is the Fourier series of a square integrable function.A basic factabout circularAbel summabilityimplies that the functionin question must be F [20, Cor. 2.15, p. 256]. In short {CM/IMI21 are the Fourier coefficientsof the square integrablefunction F(X). The firstmajor obstacle is thatin distinctionto the one dimensionalcase, it is quite difficultto demonstratethe continuityof F.
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