Summability and speed of convergence in an ergodic theorem

Given an irrational vector α in Rd, a continuous function f (x) on the torus T d and suitable weights Φ(N, n) such that ∑+∞ n=−∞ Φ(N, n) = 1, we estimate the speed of convergence to the integral ∫ T d f (y)dy of the weighted sum ∑+∞ n=−∞ Φ(N, n)f (x + nα) as N → +∞. Whereas for the arithmetic means N −1 ∑N n=1 f (x + nα) the speed of convergence is never faster than cN −1, for other means such speed can be accelerated. We estimate the speed of convergence in two theorems with different flavor. The first result is a metric one, and it provides an estimate of the speed of convergence in terms of the Fourier transform of the weights Φ(N, n) and the smoothness of the function f (x) which holds for almost every α. The second result is a deterministic one, and the speed of convergence is estimated also in terms of the Diophantine properties of the given irrational vector α ∈ Rd.