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<p>This note concerns the rapid uniform convergence of Cesàro weighted Birkhoff averages via irrational rotations on tori under certain conditions, involving arbitrary polynomial and exponential convergence rates. We discuss both finite dimensional and infinite dimensional cases, and give Diophantine rotations as examples. These provide the universality of rapid convergence for Cesàro weighted type, which is quite different from <i>L</i><sup><i>p</i></sup> (<i>p</i> &gt; 1) convergence for the unweighted one. We also show a certain optimality about our convergence rate. Besides, we introduce a multimodal weighted approach to adapt to the data sparsity, which still preserves exponential convergence.</p>

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A Note on Exponential Convergence of Cesàro Weighted Birkhoff Average and Multimodal Weighted Approach

  • 期刊论文

  • 发表日期：

    2025年9月15日

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  • Zhicheng Tong,
  • Yong Li

摘要

This note concerns the rapid uniform convergence of Cesàro weighted Birkhoff averages via irrational rotations on tori under certain conditions, involving arbitrary polynomial and exponential convergence rates. We discuss both finite dimensional and infinite dimensional cases, and give Diophantine rotations as examples. These provide the universality of rapid convergence for Cesàro weighted type, which is quite different from Lp (p > 1) convergence for the unweighted one. We also show a certain optimality about our convergence rate. Besides, we introduce a multimodal weighted approach to adapt to the data sparsity, which still preserves exponential convergence.