Probabilistic proof of a theorem of A. O. Gel'fond

In this paper we present a proof of a theorem of A. O. Gel'fond on the approximation of a continuous function by its generalized Bern?teĭn polynomials, our proof being based on probability theory.
Our proof consists in interpreting a generalized Bern?teĭn polynomial as the mathematical expectation of a function of a specially chosen random variable, followed by the use of elementary theorems of probability theory. In view of the fact that in the new proof we give no direct estimate of the function's deviation from its polynomials, the traditional arguments of the constructive theory of functions are markedly simplified.