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Estimates for coefficients in Jacobi series for functions with limited regularity by fractional calculus

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Abstract

In this paper, optimal estimates on the decaying rates of Jacobi expansion coefficients are obtained by fractional calculus for functions with algebraic and logarithmic singularities. This is inspired by the fact that integer-order derivatives fail to deal with singularity of fractional-type, while fractional calculus can. To this end, we first introduce new fractional Sobolev spaces defined as the range of the \(L^p\)-space under the Riemann-Liouville fractional integral. The connection between these new spaces and classical fractional-order Sobolev spaces is then elucidated. Under this framework, the optimal decaying rate of Jacobi expansion coefficients is obtained, based on which the projection errors under different norms are given. This work is expected to introduce fractional calculus into traditional fields in approximation theory and to explore the possibility in solving classical problems by this ‘new’ tool.

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The code used in this work will be made available upon request to the authors.

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Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments and suggestions for improvement of this paper.

Funding

This research was supported in part by the National Natural Science Foundation of China (Nos. 12201418, 12001280, 12271128 and 11971131) and the Natural Science Foundation of Heilongjiang Province of China (No. YQ2023A002).

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Authors and Affiliations

  1. School of Mathematics, Nanjing Audit University, Nanjing, 211815, People’s Republic of China

    Guidong Liu

  2. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Wenjie Liu

  3. Faculty of Computational Mathematics and Cybernetics, Shenzhen MSU-BIT University, Shenzhen, 518172, People’s Republic of China

    Beiping Duan

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  1. Guidong Liu
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  2. Wenjie Liu
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Correspondence to Beiping Duan.

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Communicated by: Bangti Jin

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Liu, G., Liu, W. & Duan, B. Estimates for coefficients in Jacobi series for functions with limited regularity by fractional calculus. Adv Comput Math 50, 68 (2024). https://doi.org/10.1007/s10444-024-10159-y

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Keywords

Mathematics Subject Classification (2010)