A sharp estimate for the rate of convergence in mean of Birkhoff sums for some classes of periodic differentiable functions

Abstract

For a badly approximable vector α, we obtain a sharp estimate for the rate of convergence in the space Lp (0 < p < ∞) of the Birkhoff means \(\frac{1}{n}\sum\nolimits_{s = 0}^{n = 1} {f(x + s\alpha )} \) for an absolutely continuous periodic function f and for functions in spaces of Bessel potentials.