Summary: Annals of Mathematics, 151 (2000), 1-33
Some spherical uniqueness theorems
for multiple trigonometric series
By J. MARSHALLASH and GANG WANG*
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 d-dimensional lattice Zd, Roman letters
x, y,.. points of the d-dimensional torus Td = [-_r, r)d, (., .) inner product,
and I| d-dimensional Euclidean norm. For a multiple trigonometric series