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If \(\alpha \) is a vector in \(\mathbb {R}^{d}\) and \(f\left ( x\right ) \) a function on the torus \(\mathbb {T}^{d}\) , define \(\displaystyle \mathcal {A}_{N}f\left ( x\right ) =\dfrac {1}{2N+1}\sum _{n=-N}^{N}f\left (x+n\alpha \right ) , \) \(\displaystyle \mathcal {B}_{N}f\left ( x\right ) =\dfrac {1}{N}\sum _{n=1-N}^{N-1}\left ( 1- \frac {\left \vert n\right \vert }{N}\right ) f\left ( x+n\alpha \right ) . \) More generally, for suitable weights \(\left \{\Phi \left (N^{-1}n\right ) \right \} _{n=-\infty }^{+\infty }\) and sequences of integers \(\left \{ a\left ( n\right ) \right \} _{n=-\infty }^{+\infty }\) , define \(\displaystyle \mathcal {C}_{N}f\left ( x\right ) =\left ( \sum _{n=-\infty }^{+\infty }\Phi \left ( N^{-1}n\right ) \right ) ^{-1}\sum _{n=-\infty }^{+\infty }\Phi \left ( N^{-1}n\right ) f\left ( x+a\left ( n\right ) \alpha \right ) . \) We estimate the speed of convergence to \(\int _{\mathbb {T} ^{d}}f(x)dx\) of these averages.

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Speed of Convergence in an Ergodic Theorem

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  • 发表日期：

    2025年

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  • Leonardo Colzani

摘要

If \(\alpha \) is a vector in \(\mathbb {R}^{d}\) and \(f\left ( x\right ) \) a function on the torus \(\mathbb {T}^{d}\) , define \(\displaystyle \mathcal {A}_{N}f\left ( x\right ) =\dfrac {1}{2N+1}\sum _{n=-N}^{N}f\left (x+n\alpha \right ) , \) \(\displaystyle \mathcal {B}_{N}f\left ( x\right ) =\dfrac {1}{N}\sum _{n=1-N}^{N-1}\left ( 1- \frac {\left \vert n\right \vert }{N}\right ) f\left ( x+n\alpha \right ) . \) More generally, for suitable weights \(\left \{\Phi \left (N^{-1}n\right ) \right \} _{n=-\infty }^{+\infty }\) and sequences of integers \(\left \{ a\left ( n\right ) \right \} _{n=-\infty }^{+\infty }\) , define \(\displaystyle \mathcal {C}_{N}f\left ( x\right ) =\left ( \sum _{n=-\infty }^{+\infty }\Phi \left ( N^{-1}n\right ) \right ) ^{-1}\sum _{n=-\infty }^{+\infty }\Phi \left ( N^{-1}n\right ) f\left ( x+a\left ( n\right ) \alpha \right ) . \) We estimate the speed of convergence to \(\int _{\mathbb {T} ^{d}}f(x)dx\) of these averages.