double Fourier series; $L^1$-convergence; logarithm bound variation double sequences
We extend the results of paper of F. Móricz (2010), where necessary conditions were given for the $L^1$-convergence of double Fourier series. We also give necessary and sufficient conditions for the $L^1$-convergence under appropriate assumptions.
 Belov, A. S.: Remarks on the convergence (boundedness) in the mean of partial sums of a trigonometric series
. Math. Notes 71 (2002), 739-748. English. Russian original translation from Mat. Zametki 71 2002 807-817. DOI 10.1023/A:1015860510199
| MR 1933102
| Zbl 1026.42010
 Zhou, S. P.: What condition can correctly generalize monotonicity in $L^1$-convergence of sine series?. Sci. Sin., Math. 40 (2010), 801-812 Chinese.