Uniqueness of Representation by Trigonometric Series

In 1870 Georg Cantor proved that a 2sr periodic omplex 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 THEOREM C (Cantor [5]). If, for every real number x N lim E ce inx = 0, N-?oo n=-N then all the complex numbers cn, 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 2iT periodic omplex valued function of a real variable coincides with the values of at most one trigonometric series. (Proof: Suppose Yaneinx = Ybeeinx for all x. Form the difference series E(an bn)einx and apply Cantor's theorem.) This theorem is remarkable on two counts. Cantor's formulation fthe problem in such a clear, decisive manner was a major mathematical event, given the point of view prevailing among his contemporaries.2 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 := ECneinx, the formal second integral, namely, F(x) = En,O(cn/(in)2)einX + 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 ofmultiple trigonometric series and to the other referee for making careful corrections and adding some historical remarks. 2In the eighteenth century, physicists just "did" Fourier series (often quite successfully) without worrying about convergence v ry much at all. When doubts about convergence b gan to arise in the nineteenth century, the first attempts atrigor were rather heavy handed. See Dauben [9, pp. 6-31] for an interesting aiscussion of the historical context.