 
Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 354, Number 12, Pages 47694788
S 00029947(02)030866
Article electronically published on July 25, 2002
SETS OF UNIQUENESS FOR SPHERICALLY CONVERGENT
MULTIPLE TRIGONOMETRIC SERIES
J. MARSHALL ASH AND GANG WANG
Abstract. A subset E of the ddimensional torus Td is called a set of unique
ness, or Uset, if every multiple trigonometric series spherically converging to 0
outside E vanishes identically. We show that all countable sets are Usets and
also that HJ sets are Usets for every J. In particular, C × Td1, where C is
the Cantor set, is an H1 set and hence a Uset. We will say that E is a UAset
if every multiple trigonometric series spherically Abel summable to 0 outside
E and having certain growth restrictions on its coefficients vanishes identically.
The abovementioned results hold also for UA sets. In addition, every UAset
has measure 0, and a countable union of closed UAsets is a UAset.
1. Introduction and Main Results
A subset E of the ddimensional torus Td
is called a set of uniqueness, or Uset,
