One-dimensional Fourier series of a function of many variables

For illustration we consider the well known example of Ch. Fefferman's function F(x,y) whose double trigonometric Fourier series S[F] diverges everywhere in the sense of Prinsheim. Namely, we establish the simultaneous convergence of the one-dimensional Fourier series S[F]1 and S[F]2 at almost all points (x,y)∈[−π,π]2 to the values F(x,y).