Geodesic
On sufficient conditions for the convergence of double series over rectangles
Matematičeskie zametki, Tome 15 (1974) no. 6, pp. 835-838.

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We prove convergence almost everywhere on $[0,2\pi]\times[0,2\pi]$ of the double Fourier series of functions $f(x,y)$ with modulus of continuity $$ \omega(f,\delta)=O\biggl(\frac1{\bigl(\ln\frac1\delta\bigr)^{1+\varepsilon}}\biggr) $$ for $\varepsilon>0$. 
@article{MZM_1974_15_6_a0,
     author = {M. Bakhbukh},
     title = {On sufficient conditions for the convergence of double series over rectangles},
     journal = {Matemati\v{c}eskie zametki},
     pages = {835--838},
     publisher = {mathdoc},
     volume = {15},
     number = {6},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_6_a0/}
}
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M. Bakhbukh. On sufficient conditions for the convergence of double series over rectangles. Matematičeskie zametki, Tome 15 (1974) no. 6, pp. 835-838. http://geodesic.mathdoc.fr/item/MZM_1974_15_6_a0/