On the Lebesgue summability of double trigonometric series

We recall that the Lebesgue summability of a single trigonometric series is defined in terms of the symmetric differentiability of the sum of the formally integrated trigonometric series in question. In this paper, we present another proof of the theorem given in Zygmund's monograph. Then we define the notion of Lebesgue summability of a double trigonometric series and extend the theorem of Fatou and Zygmund from single to double trigonometric series.