On the almost everywhere convergence of double Fourier series of integrable functions (Q1114107)

The author shows that a theorem of Menshov on the representation of functions by trigonometric series is carried over in the same form to double trigonometric series. In fact, he proves the following theorem: Let \(Q_ k\subset [0,2\pi)\), \(k=1,2\), be two perfect and nowhere dense sets. Then for each function \(F(x,y)\in L^ 1(T)\), \(T=[0,2\pi)^ 2\), there is a function \(G(x,y)\in L^ 1(T)\) such that \(G(x,y)=F(x,y)\) on \(Q=Q_ 1\times Q_ 2\) and both the rectangular and the spherical partial sums of the double Fourier series of G converge to it almost everywhere.