RECTANGULAR AND ITERATED CONVERGENCE OF MULTIPLE TRIGONOMETRIC SERIES

Abstract

In this paper we present a proof of SH. T. Tetunashvili that shows that a multiple trigonometric series that converges rectangularly everywhere actually converges iteratively everywhere to the same function. This method then solves a uniqueness problem, namely, that if a multiple trigonometric series converges rectangularly everywhere to zero, then all the coefficients are zero. We give a detailed proof in two dimensions. The result for higher dimensions may then be obtained inductively using the same proof.