 
Summary: Uniqueness for spherically convergent multiple
trigonometric series
J. Marshall Ash
Mathematics Department, DePaul University, Chicago, IL 60614
June 15, 2000
Abstract
In 1870 Cantor proved that representation of a function of one variable
by a trigonometric series can be done in only one way. In 1996 Bourgain
proved the same thing for spherical convergence and multiple trigonomet
ric series. His proof involves injecting a lot of new ideas into the theory of
uniqueness. We give here an exposition of Bourgain's proof, specialized to
the case of dimension 2. Our exposition includes a fairly general method
for finding maximal elements without resorting to the Axiom of Choice.
1 Background
The first major question that arose in the history of Fourier series was this.
Determine which functions mapping the interval [0, 2) = T into the complex
numbers can be represented in the form
S(x) =
a0
2
