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A sharp estimate for the rate of convergence in mean of Birkhoff sums for some classes of periodic differentiable functions. (English. Russian original) Zbl 1111.37002
Funct. Anal. Appl. 40, No. 1, 34-41 (2006); translation from Funkts. Anal. Prilozh. 40, No. 1, 43-51 (2006).
For each vector \((\alpha_1,\alpha_2,\ldots, \alpha_d)\in\mathbb{R}^d\) one defines the shift \(T_\alpha:\mathbb{T}^d\to\mathbb{T}^d\) on the torus \(\mathbb{T}^d\), \(T_\alpha(x)=x+\alpha\,\,(\text{mod}\,\,1)\), i.e., each coordinate of \(x+\alpha\) is computed modulo 1. The Birkhoff sum of order \(n\) over the transformation \(T_\alpha\), \(S_n^\alpha(f)\), associated to a Lebesgue integrable function \(f:\mathbb{T}^d\to\mathbb{R}\) is defined by
\[ S_n^\alpha(f)(x)=\sum_{s=0}^{n-1}f\circ T_\alpha^s=\sum_{s=0}^{n-1}f(x+s\alpha). \] The Birkhoff means \({{1}\over{n}}S_n^\alpha(f)\) converge uniformly to the spatial mean \(I(f)=\int_{\mathbb{T}^d}f(x)\, dx\) for each continuous function on \(\mathbb{T}^d\) iff \(\alpha\) is irrational (i.e., its coordinates are independent over \(\mathbb{Z}\)). For each function \(f\in L_p(\mathbb{T}^d)\), \(0<p<\infty\), the Birkhoff means converge to \(I(f)\) in \(L_p\).
The author studies the rate of convergence in the space \(L_p\) of the Birkhoff means. Namely, one obtains a sharp estimate on this rate of convergence in the case when \(\alpha\) is badly approximable and \(f\) is an absolutely continuous periodic function of zero mean or a function in the space of the Bessel potentials.
MSC:
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37A30 Ergodic theorems, spectral theory, Markov operators
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