Necessary conditions for \(L^{1}\)-convergence of double Fourier series (Q1046506)

Let \(\sum c_k \exp (ikx) \) denotes the Fourier series of a \(2 \pi\) periodic \(L^1\) integrable function. \textit{A. S. Belov} [J. Math. Sci. New York 155, No. 1, 5--17 (2008; Zbl 1154.42001) and Math. Notes 71, No. 6, 739--748 (2002); translation from Mat. Zametki 71, No. 6, 807--817 (2002; Zbl 1026.42010)] has obtained necessary conditions for the \(L^1\) convergence of the Fourier series. In particular, it is proved that the average of the absolute values of the Fourier coefficients \(c_k\) for \(m \leq |k| \leq 2m\) decays faster than ln \(m\) as \(m\) tends to infinity. The author extends the foregoing results for double Fourier series.