The Fejer sums for periodical measures and the norms of the deviations from the limit in the von Neumann ergodic theorem both are calculating, in fact, with the same formulas (by integrating of the Fejer kernels) — and so, this ergodic theorem, in fact, is a statement about the asymptotic of the growth of the Fejer sums at zero point of the spectral measure of corresponding dynamical system. It gives a possibility to rework well-known estimates of the rates of convergence in the von Neumann ergodic theorem into the estimates of the Fejer sums in the point for periodical measures — for example, we obtain natural criteria of polynomial growth and polynomial degree of these sums. And vice versa, numerous in the literature estimates of the deviations of Fejer sums in the point allow to obtain new estimates of the rate of convergence in this ergodic theorem.