Geodesic
Approximation of periodic functions of several variables by bilinear forms
Izvestiya. Mathematics , Tome 28 (1987) no. 1, pp. 133-150

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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. Bibliography: 16 titles. 
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     author = {V. N. Temlyakov},
     title = {Approximation of periodic functions of several variables by bilinear forms},
     journal = {Izvestiya. Mathematics },
     pages = {133--150},
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     volume = {28},
     number = {1},
     year = {1987},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a6/}
}
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V. N. Temlyakov. Approximation of periodic functions of several variables by bilinear forms. Izvestiya. Mathematics , Tome 28 (1987) no. 1, pp. 133-150. http://geodesic.mathdoc.fr/item/IM2_1987_28_1_a6/