 
Summary: Annals of Mathematics, 151 (2000), 133
Some spherical uniqueness theorems
for multiple trigonometric series
By J. MARSHALLASH and GANG WANG*
Abstract
We prove that if a multiple trigonometric series is spherically Abel sum
mable everywhereto an everywherefinite function f(x) which is bounded below
by an integrable function, then the series is the Fourier series of f(x) if the
coefficients of the multiple trigonometric series satisfy a mild growth condition.
As a consequence, we show that if a multiple trigonometric series is spherically
convergent everywhere to an everywhere finite integrable function f(x), then
the series is the Fourier series of f(x). We also show that a singleton is a
set of uniqueness. These results are generalizations of a recent theorem of J.
Bourgain and some results of V. Shapiro.
1. Introduction and summary of results
We start with the question of spherical uniqueness of multiple trigono
metric series for integrable functions under Abel summability. Greek let
ters ,, ,.. will denote points of the ddimensional lattice Zd, Roman letters
x, y,.. points of the ddimensional torus Td = [_r, r)d, (., .) inner product,
and I ddimensional Euclidean norm. For a multiple trigonometric series
