Uniqueness of double Fourier-Stieltjes series (Q1347775)

The main result of this paper is its Theorem 2, i.e.: Let \(F\) be a function of bounded variation over \([0,2\pi]\times [0,2\pi]\) in the sense of Hardy and Krause. The Fourier-Stieltjes series of \(F\) vanishes identically if and only if the following four conditions are satisfied: (i) \(F(2\pi,2\pi)-F(0,2\pi)-F(2\pi,0)+F(0,0)=0,\) (ii)\(F(\cdot,2\pi)\) and \(F(\cdot,0)\) differ only by a constant on \([0,2\pi]\), except possibly at the points of a countable set. (iii)\(F(2\pi,\cdot)\) and \(F(0,\cdot)\) differ only by a constant on \([0,2\pi]\), except possibly at the points of a countable set. (iv) \(F(x,y)\) can be represented as \(g(x)+h(y)\) on \([0,2\pi]\times [0,2\pi]\), except possibly at the points on a countable number of horizontal and vertical line segments, where \(g\) and \(h\) are functions of bounded variation on \([0,2\pi]\). This result is an extension of the well-known one dimensional result.