Bilinear and trigonometric approximations of periodic functions of several variables of Besov classes $B_{p, \theta}^r$

We obtain order-sharp estimates for bilinear approximations of periodic functions of $2d$ variables of the form $f(x,y)=f(x-y)$, $x, y\in \pi_d = \prod_{j=1}^d[-\pi, \pi]$, obtained from functions $f(x)\in B_{p, \theta}^r$, $1\le p<\infty$, by translating the argument $x\in \pi_d$ by vectors $y\in \pi_d$. We also study the deviations of step hyperbolic Fourier sums on the classes $B_{1, \theta}^r$ and the best orthogonal trigonometric approximations in $L_q$, $ 1<q<\infty$, of functions belonging to these classes.