We present necessary and sufficient conditions for double sine, sinecosine, cosine-sine and double cosine series in terms
of coefficients that their sums belong to double Lipschitz classes. Some classical results on single trigonometric series
and some new results of Fülöp  on double trigonometric series are extended.
We construct a new kind of rational operator which can be used to approximate functions with endpoints singularities by algebric weights in [−1,1], and establish new direct and converse results involving higher modulus of smoothness and a very general class of step functions, which cannot be obtained by weighted polynomial approximation. Our results also improve related results of Della Vecchia .
In this paper, we generalize two important results of Bagota and Móricz , and generalize our earlier results in  from
one-variable to two-variable case. As special applications, we prove that the generalized jump of f(x, y) at some point (x0, y0) can be determined by the higher order mixed partial derivatives of the Abel-Poisson mean of double Fourier series and the
higher order mixed partial derivatives of the Abel-Poisson means of the three conjugate double Fourier series.
We introduce a new kind of double sequences named MVBVDS and some new classes of weight functions to study the weighted integrability
of the double trigonometric series. Several results of Chen, Marzuq, Móricz, Ram and Singh Bhatia (see –) are generalized and some new results are established.