Estimates of the best bilinear approximations for the classes of $(ψ,β)$-differentiable periodic multivariate functions

Abstract

Order estimates are obtained for the best bilinear approximations of $2d$-variable functions $f(x y),\; x, y \in \pi_d,\; \pi_d =\prod^d_{j=1} [ \pi , \pi ]$, formed by $d$-variable functions $f(x) \in L^{\psi}_{\beta} ,p$ by the shifts of their argument $x \in \pi_d$ by all possible values of $y \in \pi_d$ in the space $L_{q_1,q_2} (\pi 2d)$. The results include various relations between the parameters $p, q_1$ and $q_2$.