Uniqueness of Representation by Trigonometric Series

In 1870 Georg Cantor proved that a 2s periodic complex valued function of a real variable coincides with the values of at most one trigonometric series. We present his proof and then survey some of the many one dimensional generalizations and extensions of Cantor's theorem. We also survey the situation in higher dimensions, where a great deal less is known. 1. Cantor's uniqueness theorem. In 1870 Cantor proved THEOREMC (Cantor [5]). If, for evely real number x N lim cneinX= 0, N-rw " = N then all the complex numbers c,, n = 0,1, 1,2, 2,. . . are zero. This is called a uniqueness theorem because it has as an immediate corollary the fact that a 27r periodic complex valued function of a real variable coincides with the values of at most one trigonometric series. (Proof: Suppose Cane1"" = CbnelnXfor all x. Form the difference series C(a, bn)el"" and apply Cantor's theorem.) This theorem is remarkable on two counts. Cantor's formulation of the problem in such a clear, decisive manner was a major mathematical event, given the point of view prevailing among his c~ntemporaries.~ Equally enjoyable to behold is the rapid resolution that we will now sketch. Cantor's theorem is relatively easy to prove, if, as Cantor did, you have studied Riemann's brilliant idea of associating to a general trigonometric series T := Ccnei"", the formal second integral, namely, F(x) := C, ,o(cn/(in)2)ei"x+ co(x2/2). For some interesting remarks on the importance of this idea, see the very enjoyable survey article of Zygmund [27]. Define the second Schwarz derivative D of a ''The research presented here was supported in part by a grant from the University Research Council of DePaul University. The author is grateful to one referee for proposing an expanded treatment of multiple trigonometric series and to the other referee for making careful corrections and adding some historical remarks. 2 ~ n the eighteenth century, physicists just "did" Fourier series (often quite successfully) without worrying about convergence very much at all. When doubts about convergence began to arise in the nineteenth century, the first attempts at rigor were rather heavy handed. See Dauben [9,pp. 6-31] for an interesting discussion of the historical context. J. MARSHALL ASH [December function G(x) by G(x + h) 2G(x) + G(x h) DG(x) := lim h-0 h2 The steps of the proof are: 1. Since T converges everywhere, it is immediate that for every value of x, cneinX+ c-,e-'"" + 0 as n -+ oo.By the Cantor-Lebesgue theorem, Icnl + Ic-,l + 0 as n + oo. Appendix 1 gives Cantor's weak but easy version of this. (For a statement of the more powerful Cantor-Lebesgue theorem see the survey article by Roger Cooke [8]. The proof given there is much shorter than the one in Appendix 1, but requires some of the machinery of modern analysis.) 2. By the Weierstrass M-test from cneinX inx sup(IcnI + Ic-,I) n2 it follows that F(x) c0(x2/2) is a continuous function and that F is the uniform limit of its partial sums. (See Theorems 25.7 and 24.3 in [14] for the M-test.) 3. An important result of Schwarz's states that if G is continuous and DG(x) = 0 for all x, then G is a linear function. (See [5,pp. 82-83].) Before presenting a proof of this, Cantor remarks that Schwarz mailed this result to him from Ziirich.) We give a proof in Appendix 2. 4. Since ' h ' 2 e i ( x + h ) 2 e i ~+ e i (x-h) sin = e ' ~ -2 h2 -h \ 2 (Check this.), we have I nh 2 F ( x + h) 2F(x) + F ( x h) = cO+ x cneinX-sin 2 ' h' nh ' n + O , 2 , From a, + Cy'lan = 0 it follows that lim,,, a, + CF='=,a,(sin nk/nk)2 = 0. See Appendix 3 for Riemann's summation by parts proof of this. Hence DF(x) = 0 so F(x) = ax + /3 for some a and /3. 5. The right side of the equation xL co+ ax + /3 = x l e i n x 2 (in)' is bounded. (From 2. above it is continuous, hence bounded on [O,2a], and hence bounded everywhere by periodicity.) Letting x -+ oo twice first shows c, = 0 and then a = 0. 6. From the observation made in step 2 above, we see that the sequence 875 19891 UNIQUENESS OF REPRESENTATION BY TRIGONOMETRIC SERIES converges uniformly to 0. But for each n + 0, (in)' i2;, c, = ( x ) e ~ dx 277 for all N 2 n by the orthonormality of {elnx/ fi), so letting N + co gives c, = 0 for all n # 0. (The uniformity of convergence allows the interchange of limit and integral.) Q.E.D. Cantor's beautiful theorem suggests a variety of extensions and generalizations. The remaining four sections of this paper will consider some of these. 2. Summability and uniqueness. Can we improve Theorem C by weakening the assumption of convergence to zero to an assumption of being merely summable to zero? As is often the case in mathematics, the starting point is a counterexample which destroys the "obvious extension." The trigonometric series Ccneinx is said to be Abel summable to s if for each r, 0 < r < 1, f(x, r ) := Ccneinxrlnl converges and if lim,,,f(x, r ) = s. Let z := reix = r(cos x + i sin x). Differentiate the identity 00 00 1 1 5 1rcosx (cosnx)rn =a(x =a(-. -) = n = ~ n = ~ 1 z 1 5 1 2 r c o s x + r 2 with respect to x to obtain a3 (1 r 2 ) r s i n x C (n sin nx)rn = n=l (I 2rcosx + r 2 1 2 ' If x # 0, as r + I-, the right side tends to 0; while if x = 0, every term of f (0, r ) is 0, SO that f(0, r ) = 0, whence lim,,,f(0, r ) = 0. Thus C?=,n sin nx is everywhere Abel summable to 0. Although this example is unpleasant, it turns out to be just about the worst thing that can happen. THEOREM [25, VOL. I, PP. 352, 3831, [22], [23]). If cn/lnl + 0 as V (VERBLUNSKY In 1 + co and Ccneinx is Abel summable to 0 at every x, then all c, are 0. 3. Higher dimensions. When we move from one to several dimensions, the picture becomes much more cloudy. Here is the land of opportunity. Almost nothing is known; almost every question that the novice might ask turns out to be an open question. The first goal is to mimic Cantor's Theorem C. Even this apparently modest goal remains to a large extent unachieved. The hypothesis that Ccneinx converges to 0 has several different interpretations in higher dimensions. Most of these can be illustrated in two dimensions, so to ease notation I will restrict myself to that case. The basic object will be the double trigonometric series T(x, y ) := C(m,,) We define a rectangular partial sum of T to be Z~mnei(mx+ny). a diamond shaped partial sum to be J. MARSHALL ASH [December and a circular partial sum to be We freeze x and y and make five different definitions of convergence. If T, -+ s as r + co say T is circularly convergent to s. If T n + s as r + co say T is triangularly convergent to s. If Tnn + s as r + co say T is square convergent to s. If T, + s as min{m, n ) -+ co say T is unrestrictedly rectangularly convergent to s. If Tmn + s as min{m, n ) + co in such a way that m/n and n/m stay less than e, and if this happens for each (arbitrarily large) e > 1, say T is restrictedly rectangularly convergent to s. To each of these five notions of convergence there corresponds a putative extension of Theorem C. The first of these to be proved was the following. THEOREMSC (V. Shapiro and R. Cooke [19], [7]). If T(x, y) is circularly convergent to 0 everywhere, then all c,, are 0. In 1957 Victor Shapiro proved a two-dimensional version of Theorem V which implied a weak version of Theorem SC that required the additional hypothesis that 1 C IC,,I -+ o as r -+ co. (1) ( r 1 ) 2 < m 2 + n 2 ~ r 2 That this hypothesis was not needed was a consequence of a generalization of the Cantor-Lebesgue theorem due to Roger Cooke in 1971. (See Theorem A1 below. A survey of Cooke's theorem and extensions of it by Zygmund [26] and Connes [6] can be found in the MONTHLY article by Cooke [a].) The proof of Theorem SC is modeled after Verblunsky's Theorem V mentioned above. Both these theorems carry out the bemann-Cantor program of integrating twice and then differentiating twice. To get at the ideas behind Theorem SC, assume that T is circularly convergent to zero everywhere. Write M := (m, n), X := (x, y), M . X = mx f ny, and add the assumption that c, = 0. (This simplifies the notation, but not the proof.) Let