A Survey of Multidimensional Generalizations of Cantor’s Uniqueness Theorem for Trigonometric Series

Abstract

Georg Cantor’s pointwise uniqueness theorem for one-dimensional trigonometric series says that if, for each x in [0,2π), ∑c _n e ^inx=0, then all c _n =0. In dimension d, d≥2, we begin by assuming that for each x in [0,2π)^d, ∑c _n e ^inx=0 where $n = ({n}_{1},\ldots,{n}_{d})$ and $nx = {n}_{1}{x}_{1} + \cdots+ {n}_{d}{x}_{d}$. It is quite natural to group together all terms whose indices differ only by signs. But here there are still several different natural interpretations of the infinite multiple sum, and, correspondingly, several different potential generalizations of Cantor’s theorem. For example, in two dimensions, two natural methods of convergence are circular convergence and square convergence. In the former case, the generalization is true, and this has been known since 1971. In the latter case, this is still an open question. In this historical survey, I will discuss these two cases as well as the cases of iterated convergence, unrestricted rectangular convergence, restricted rectangular convergence, and simplex convergence.