We investigated the Bernstein function $φ: \mathbb{R} → \mathbb{R}, φ(λ) = E_{n-1} (f - r_n + λe_n),$ set its basic properties and found an explicit form in the case when $f(x) = x^2, n=1.$

[1] Bernstein S.N. Sur le probléme inverse de la theorie de la meilleure approximation des fonctions continues // Comp. Rend. - 1938. - 206 . - P. 1520-1523.

[2] Kroo A. The continuoty of best approximatious. Acta Math. Acad. Sci. Hungary. - 1977. - 30  .- P.175- 188.

[3] Akhiezer N.I. Lectures on approximation theory. - M.: Nauka, 1965. - 407 p.

[4] Bernstein S.N. On an inverse problem of approximation theory / /Collected works in 4 volumes. Moscow: Publishing house of the USSR Academy of Sciences, 1954. - V.2. - P.292-294.

[5] Voloshyn G.A., Maslyuchenko V.K. On the functional generalization of one of Bernstein's theorems //International conference dedicated to the 100th anniversary of M.M. Bogolyubov and the 70th anniversary of M.I. Nagnibida., June 8-13, 2009. Abstracts of reports. Chernitsi 2009. - P.149-150.

[6] Voloshin G.A., Maslyuchenko V.K. On the question of generalization of one Bernstein theorem/ / FM 2009 Conference "Functional Methods in Approximation Theory and Operator Theory III"dedicated to the memory of V.K. Dzyadyk (1919-1998)., August 22-26, 2009. Abstracts. Kuiv, 2009. - P.106-107.