Convergence of multiple Fourier series for functions of bounded variation (Q1274026)

The pointwise convergence in Pringsheim's sense of multiple Fourier series of functions of bounded variation was proved by G. H. Hardy in 1906. In the present paper, the pointwise convergence of the partial sums of Fourier series is considered over a given sequence of bounded sets in the space of harmonics. Sufficient conditions are proved for the convergence, which are also necessary in the particular case when these sets are convex with respect to each coordinate direction.