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Keywords:
double Fourier series; $L^1$-convergence; logarithm bound variation double sequences
Summary:
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.
References:
[1] 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
[2] He, J. L., Zhou, S. P.: On $L^1$-convergence of double sine series. Acta Math. Hung. 143 (2014), 107-118. DOI 10.1007/s10474-013-0376-y | MR 3215608 | Zbl 1324.42011
[3] Kaur, K., Bhatia, S. S., Ram, B.: $L^1$-convergence of complex double Fourier series. Proc. Indian Acad. Sci., Math. Sci. 113 (2003), 355-363. DOI 10.1007/BF02829630 | MR 2020071 | Zbl 1041.42005
[4] Móricz, F.: Necessary conditions for $L^1$-convergence of double Fourier series. J. Math. Anal. Appl. 363 (2010), 559-568. DOI 10.1016/j.jmaa.2009.09.030 | MR 2564875 | Zbl 1182.42009
[5] Tikhonov, S.: On $L_1$-convergence of Fourier series. J. Math. Anal. Appl. 347 (2008), 416-427. DOI 10.1016/j.jmaa.2008.05.048 | MR 2440338 | Zbl 1257.42009
[6] Zhou, S. P.: What condition can correctly generalize monotonicity in $L^1$-convergence of sine series?. Sci. Sin., Math. 40 (2010), 801-812 Chinese.
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