New Uniqueness Theorems for Trigonometric Series

A uniqueness theorem is proved for trigonometric series and another one is proved for multiple trigonometric series. A corollary of the second theorem asserts that there are two subsets of the d-dimensional torus, the first having a countable number of points and the second having 2d points such that whenever a multiple trigonometric series "converges" to zero at each point of the former set and also converges absolutely at each point of the latter set, then that series must have every coefficient equal to zero. This result remains true if "converges" is interpreted as any of the usual modes of convergence, for example as "square converges" or as "spherically converges."