Best bilinear approximations of functions from Nikol’skii–Besov classes

We obtain exact-order estimates for the best bilinear approximations of Nikol’skii–Besov classes in the functional spaces Lq. 2d /: In the present paper, we investigate the approximation of functions of 2d variables from the Nikol’skii–Besov classes B p; by linear combinations of products of functions of d variables. Approximations of this type are called bilinear, and, along with the classical approximation methods using algebraic or trigonometric polynomials, they play an important role in approximation theory (for more details, see the next section). First, we give the definitions of the classes of functions and approximation characteristic considered in this paper. Let R ; d 1; denote the d -dimensional Euclidean space with elements x D .x1; : : : ; xd / and let Lp. d / be the space of functions f that are 2 -periodic in each variable and such that kf kp D @.2 / d Z