Victor Shapiro and the Theory of Uniqueness for Multiple Trigonometric Series

Abstract

In 1870, Georg Cantor proved that if a trigonometric series converges to 0 everywhere, then all its coefficients must be 0. In the twentieth century this result was extended to higher dimensional trigonometric series when the mode of convergence is taken to be spherical convergence and also when it is taken to be unrestricted rectangular convergence. We will describe the path to each result. An important part of the first path was Victor Shapiro’s seminal 1957 paper, Uniqueness of multiple trigonometric series. This paper also was an unexpected part of the second path.