The order of growth of quadratic partial sums of a double Fourier series (Q1316877)

It is well-known that the \(n\)th partial sum of a one-dimensional Fourier series increases at most as \(o(\log n)\) at the Lebesgue points. The authoress studies the analogous problem for two-dimensional Fourier series. To this effect, she introduces the notion of an \(A\)-Lebesgue point with respect to a periodic function of two variables, where \(A = \{A_ j : j = 0,1,\dots\}\) is a sequence of positive numbers, and estimates the order of magnitude of the \(n\)th quadratic partial sum of a two-dimensional Fourier series at the \(A\)-Lebesgue points. She also shows that these estimates are the best possible.