Ergodic averages with deterministic weights
Ergodic averages with deterministic weights
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具有确定性权重的遍历平均值
DOI：
10.5802/aif.1894
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
Dominique Schneider
中科院分区：
文献类型：
--
作者：
F. Durand;Dominique Schneider
The purpose of this paper is to study ergodic averages with deterministic weights. More precisely we study the convergence of the ergodic averages of the type $\frac{1}{N} \sum_{k=0}^{N-1} \theta (k) f \circ T^{u_k}$ where $\theta = (\theta (k) ; k\in \NN)$ is a bounded sequence and $u = (u_k ; k\in \NN)$ a strictly increasing sequence of integers such that for some $\delta<1$ $$ S_N (\theta, u) := \sup_{\alpha \in \pRR} | \sum_{k=0}^{N-1} \theta (k) \exp (2i\pi \alpha u_k) | = O (N^{\delta}) \leqno{({\cal H}_1)} $$ i.e., there exists a constant $C$ such that $S_N (\theta, u) \leq C N^{\delta} $. We define $\delta (\theta, u)$ to be the infimum of the $\delta $ satisfying $\H_1$ for $\theta $ and $u$.
The purpose of this paper is to study ergodic averages with deterministic weights. More precisely we study the convergence of the ergodic averages of the type $\frac{1}{N} \sum_{k=0}^{N-1} \theta (k) f \circ T^{u_k}$ where $\theta = (\theta (k) ; k\in \NN)$ is a bounded sequence and $u = (u_k ; k\in \NN)$ a strictly increasing sequence of integers such that for some $\delta<1$ $$ S_N (\theta, u) := \sup_{\alpha \in \pRR} | \sum_{k=0}^{N-1} \theta (k) \exp (2i\pi \alpha u_k) | = O (N^{\delta}) \leqno{({\cal H}_1)} $$ i.e., there exists a constant $C$ such that $S_N (\theta, u) \leq C N^{\delta} $. We define $\delta (\theta, u)$ to be the infimum of the $\delta $ satisfying $\H_1$ for $\theta $ and $u$.