On L 1-convergence of double sine series

Necessary and sufficient conditions for the $L^1$-convergence of double Fourier series

“…Before we prove Theorem 2.3, we need an inequality. A similar inequality was proved in [2], Lemma 2, although we think their proof is incomplete and we hereby give a complete one.…”

“…Now we establish necessary and sufficient conditions for the convergence in L 1 -norm in case of coefficients of special type. We use the concept of logarithm bound variation double sequences, see [2]. A double sequence…”

“…satisfied for all the row and column subsequences of {c jk } with the same constant C {c jk } . Indeed, by [2], Lemma 1,…”