 
Summary: A SURVEY OF UNIQUENESS QUESTIONS IN MULTIPLE
TRIGONOMETRIC SERIES
J. MARSHALL ASH AND GANG WANG
Dedicated to Victor Shapiro.
Abstract. The issue is uniqueness of representation by multiple trigonomet
ric series. Two basic uniqueness questions, one about series which converge to
zero and the other about series which converge to an integrable function, are
asked for each of four modes of convergence: unrestricted rectangular conver
gence, spherical convergence, square convergence, and restricted rectangular
convergence. Thus there are eight basic questions in each dimension. In all
dimensions, both uniqueness theorems are valid for unrestricted rectangular
convergence, as is the ...rst uniqueness theorem for spherical convergence. The
second uniqueness theorem holds for circular convergence in dimension 2. All
the other questions are still open. The positive work will be surveyed along
with related work involving extensions of the CantorLebesgue theorem.
1. Introduction
In 1974 Marshall Ash gave a talk entitled "Multiple trigonometric series" in
which were listed only 4 substantial uniqueness theorems for convergent multiple
trigonometric series [A1]. They were
1 if a double trigonometric series is spherically convergent to zero everywhere,
