Approximation of periodic functions of several variables by bilinear forms

The orders of the quantities
$$ \tau_M(F)_{p_1,p_2}=\sup_{f\in F}\inf_{\substack{u_i(\mathbf x),v_i(\mathbf y)\\i=1,\dots,M}}\biggl\|f(\mathbf x-\mathbf y)-\sum_{i=1}^Mu_i(\mathbf x)v_i(\mathbf y)\biggr\|_{p_1,p_2} $$
are obtained, where $F$ is a class of functions with mixed derivative, or the corresponding prelimiting difference, bounded in $L_q$. In the process some results of independent interest are obtained: a generalization of the Hardy–Littlewood theorem, and the orders of the best $M$-term trigonometric approximations.
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