 
Summary: Contemporary Mathematics
Uniqueness Questions for Multiple Trigonometric Series
J. Marshall Ash and Gang Wang
Abstract. We survey some recent results on the uniqueness questions on
multiple trigonometric series. Two basic questions, one about series which
converges to zero and the other about the series which converge to an inte
grable function, are asked for four modes of convergence: unrestricted rectan
gular convergence, spherical convergence, square convergence, and restricted
rectangular convergence. We will either get into the details or outline some
of the proofs for the known uniqueness theorems. Some results on the sets
of uniqueness are also given. Finally, we will mention some interesting open
questions in this area. Some of them are even onedimensional. We assume the
reader has some basic knowledge of measure theory and Fourier analysis. Most
of the topics and materials can be understood by upper level undergraduate
students.
Contents
1. Introduction 1
2. Some CantorLebesgue Type Theorems 6
3. A Uniqueness Theorem for Unrestrictedly Rectangular Convergence 10
4. A Uniqueness Theorem for Spherical Convergence 14
