 
Summary: UNIQUENESS FOR HIGHER DIMENSIONAL TRIGONOMETRIC
SERIES
J. MARSHALL ASH
Abstract. Five uniqueness questions for multiple trigonometric series are
surveyed. If a multiple trigonometric series converges everywhere to zero in
the sense of spherical convergence, of unrestricted rectangular convergence, or
of iterated convergence, then that series must have every coefficient being zero.
But the cases of square convergence and restricted rectangular convergence
lead to open questions.
1. Introduction
Let Td
= [0, 1)d
be the d dimensional torus. This means that Td
is a bounded
part of d dimensional Euclidean space, but that addition is modulo 1 in each co
ordinate. Let {n (x)}n=1,2,... be a real or complex valued system of functions
that are in L2
Td
= f : Td
C : Td f
