Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems

We present estimates (which are necessarily spectral) of the rate of convergence in the von Neumann ergodic theorem in terms of the singularity at zero of the spectral measure of the function to be averaged with respect to the corresponding dynamical system as well as in terms of the decay rate of the correlations (i.e., the Fourier coefficients of this measure). Estimates of the rate of convergence in the Birkhoff ergodic theorem are given in terms of the rate of convergence in the von Neumann ergodic theorem as well as in terms of the decay rate of the large deviation probabilities. We give estimates of the rate of convergence in both ergodic theorems for some classes of dynamical systems popular in applications, including some well-known billiards and Anosov systems.