Summary: Uniqueness for spherically convergent multiple
J. Marshall Ash
Mathematics Department, DePaul University, Chicago, IL 60614
June 15, 2000
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.
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