Geodesic
Convergence of double series
Matematičeskie zametki, Tome 13 (1973) no. 3, pp. 341-350.

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The article considers the question of the mutual relationship of different forms of convergence of double series. When the condition $$a_{ik}=o\left(\frac1{i^2+k^2}\right)$$ is satisfied, the following are equivalent: convergence over squares, convergence over rectangles, convergence over circles. The conditions obtained cannot be strengthened. Several deductions are made relating to the convergence of double trigonometric series. 
@article{MZM_1973_13_3_a1,
     author = {M. Bakhbukh},
     title = {Convergence of double series},
     journal = {Matemati\v{c}eskie zametki},
     pages = {341--350},
     publisher = {mathdoc},
     volume = {13},
     number = {3},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_13_3_a1/}
}
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M. Bakhbukh. Convergence of double series. Matematičeskie zametki, Tome 13 (1973) no. 3, pp. 341-350. http://geodesic.mathdoc.fr/item/MZM_1973_13_3_a1/