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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
 

Summary: TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 354, Number 12, Pages 47694788
S 0002-9947(02)03086-6
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 d-dimensional torus Td is called a set of unique-
ness, or U-set, if every multiple trigonometric series spherically converging to 0
outside E vanishes identically. We show that all countable sets are U-sets and
also that HJ sets are U-sets for every J. In particular, C Td-1, where C is
the Cantor set, is an H1 set and hence a U-set. We will say that E is a UA-set
if every multiple trigonometric series spherically Abel summable to 0 outside
E and having certain growth restrictions on its coefficients vanishes identically.
The above-mentioned results hold also for UA sets. In addition, every UA-set
has measure 0, and a countable union of closed UA-sets is a UA-set.
1. Introduction and Main Results
A subset E of the d-dimensional torus Td
is called a set of uniqueness, or U-set,

  

Source: Ash, J. Marshall - Department of Mathematical Sciences, DePaul University

 

Collections: Mathematics