A Cantor-Lebesgue theorem in two dimensions

Abstract

We prove a partial extension of the classical result of Cantor and Lebesgue to two dimensions, for circular summation of trigonometric series. We assume convergence almost everywhere rather than merely on a set of positive measure. Although the theorem we prove has applications in uniqueness theorems, its purpose is to suggest what may be true in general, since the proof does not generalize to higher dimensions.