We show that the sequence of Bernstein’s operators $B_n$ of several variables is a $pu$ - approximating sequence for the subspace $P$ of all polynomials on the cube $[0,1]^S.$ Using these operators, we obtain a theorem on the layer-wise uniform approximation of separately continuous functions on $X × [0,1]^S.$

[1] Vlasiuk G.A., Masliuchenko V.K. Bernstein polynomials and non-negatively continuous functions // Scientific Bulletin of Chernivtsi University. Iss. 336-337. Mathematics - Chernivtsi: Ruta, 2007. - P. 52-59.

[2] Fikhtenholtz G.M. Fundamentals of Mathematical Analysis. Vol.2. - C.-Petersburg-Moscow-Krasnodar: Lan, 2005. - 464 p.

[3] Voloshin G.A., Maslyuchenko V.K. On the approximation of differentially continuous functions, $2π$ - periodic with respect to the second variable // Karp. math. publ. - 2010. - 2, №1 - P. 4-14.

[4] Voloshin G.A., Maslyuchenko V.K., Maslyuchenko O.V. On the approximation of individually and jointly continuous functions // Karp. math. publ. - 2010 - 2, №2 - P.11-21.

[5] Voloshyn G.A., Maslyuchenko V.K., Nesterenko O.N. On approximation of mappings with values in the space of continuous functions // Carpathian Math.

[6] Marchuk L.M. On the approximation of functions of many variables by Bernstein polynomials. Diploma thesis. - Chernivtsi, 1986. - 36 p.

[7] Maslyuchenko V.K. First types of topological vector spaces: Ruta, 2002. - 72 p.

[8] Maslyuchenko V.K. Linear continuous operators: Ruta, 2002. - 72 p.