On the Lebesgue summability of regularly convergent double trigonometric series

We recall that the Lebesgue summability of the double trigonometric series $$\sum\limits_{m \in \mathbb{Z}} {\sum\limits_{m \in \mathbb{Z}} {c_{m,n} e^{i(mx + ny)} } }$$ (*) is defined in terms of the symmetric differentiability of its formally integrated series with respect to both variables. Under conditions weaker than the known ones in the literature, in this paper we prove that if the series (*) converges regularly at a point (x, y) to the sum s, then it is also Lebesgue summable at (x, y) to s (cf. the conditions (2.6) and ((2.7) in the known Theorem 1 and the conditions (3.1) and (3.2) in our new Theorem 2). This also demonstrates the superiority of the notion of regular convergence over the notion of convergence in Pringsheim’s sense of double series of numbers (see other examples in [5]).