The article considers the  L_P(T²) Lebesque space of periodec functions of two variables. The problems of approximation of functions of two variables by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses are stydied.
Value $$E_{Q^{\gamma \:}_n}\left(f\right)_p=\:inf\:_{t\in \left(Q^{\gamma \:}_n\right)}\lceil f-t\rceil _p,\:i\le p\le \infty$$
the best approximation of the function f(x) by trigonometric polynomials with “numbers” of harmonics from a step hyperbolic cross of $$Q^{\gamma }_n$$ The article consists of two sections. The first section contains some well-known statements necessary to prove the main results. In the second section, exact estimates of the best approximations of certain functions are established. These estimates make it possible to estimate the upper bounds of the best approximations for certain classes of functions. As approximation apparatuses, trigonometric polynomials with a spector from a stepwise hyperbolic cross are used. The questions considered in this work belong to the circle of questions studied in the works of K. I. Babenko, S. A.Telyakovsky, I. S.Bugrova, N.S.Nikolsky.