Approximation by Fourier sums of classes of functions with several bounded derivatives

An ordered estimate is obtained for the approximation by Fourier sums, in the metric $\widetilde{\mathscr L}$, $q=(q_1,\dots,q_n)$, $1<q_<\infty$, $j=1,\dots,n$, of classes of periodic functions of several variables with zero means with respect to all their arguments, having $m$ mixed derivatives of order $\alpha^1,\dots,\alpha_i^m$, $\alpha^i\in R^n$. which are bounded in the metrics of$\widetilde{\mathscr L}_{p^1},\dots,\widetilde{\mathscr L}_{p^m}$, $p^i=(p_1^i,\dots,p_n^i)$, $1<p_j^i<\infty$, $i=1,\dots,m$, $j=1,\dots,n$ by the constants $\beta_1,\dots,\beta_m$, respectively.