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Local behaviors of Fourier expansions for functions of limited regularities

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Abstract

Based on the explicit formula of the pointwise error of Fourier projection approximation and by applying van der Corput-type Lemma, optimal convergence rates for periodic functions with different degrees of smoothness are established. It shows that the convergence rate enjoys a decay rate one order higher in the smooth parts than that at the singularities. In addition, it also depends on the distance from the singularities. Ample numerical experiments illustrate the perfect coincidence with the estimates.

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Acknowledgements

The authors would like to thank Desong Kong for his useful suggestions on the last numerical experiment. The authors are grateful to the anonymous referees for their valuable comments and suggestions for improvement of this paper.

Funding

This work was supported by the National Natural Science Foundation of China (No. 12271528). The first author is supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2020zzts030).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, People’s Republic of China

    Shunfeng Yang & Shuhuang Xiang

  2. School of Mathematics and Physics, Southwest Forest University, Kunming, Yunnan, 650224, People’s Republic of China

    Shunfeng Yang

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  1. Shunfeng Yang
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Correspondence to Shuhuang Xiang.

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Communicated by: Yuesheng Xu

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Yang, S., Xiang, S. Local behaviors of Fourier expansions for functions of limited regularities. Adv Comput Math 50, 47 (2024). https://doi.org/10.1007/s10444-024-10136-5

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