Convergence of double series

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.