Double cosine-sine series and Nikol'skii classes in uniform metric

In the this paper, we give neccessary and sufficient conditions for a function even with respect to the first argument but odd with respect to the second one to belong to the Nikol'skii classes defined by a mixed modulus of smoothness of a mixed derivative (both have arbitrary integer orders). These conditions involve the growth of partial sum of Fourier cosine-sine coefficients with power weights or the rate of decreasing to zero of these coefficients. A similar problem for generalized "small" Nikol'skii classes is also treated.