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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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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DOI: https://doi.org/10.1007/s10444-024-10136-5