The Fejér Integrals and the Von Neumann Ergodic Theorem with Continuous Time

Abstract

The Fejer integrals for finite measures on the real line and the norms of the deviations from the limit in the von Neumann ergodic theorem are calculating, in fact, with the same formulas (by integrating of the Fejer kernels). Thus this ergodic theorem is a statement about the asymptotic of the growth of the Fejer integrals at zero point of the spectral measure of corresponding dynamical system. It gives a possibility to rework well-known estimates of convergence rates in the von Neumann ergodic theorem into the estimates of the Fejer integrals at a point for finite measures: for example, natural criteria of polynomial growth and polynomial decay of these integrals are obtained. And vice versa, numerous known estimates of the deviations of Fejer integrals at a point allow to obtain new estimates of convergence rates in this ergodic theorem.