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Optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities

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Abstract

In this paper, new and optimal asymptotics on the decay of the coefficients for functions of limited regularity expanded in terms of Jacobi and Gegenbauer polynomial series are presented. For a class of functions with interior singularities, the decay of the coefficient is of the same asymptotic order for arbitrary \(\alpha ,\,\beta >-1\), which confirms that the decay of the coefficients in the Jacobi polynomial series without normalization is a factor of \( \sqrt{n}\) slower compared with the Chebyshev expansion. While for functions with boundary singularities, the decay depends on \(\alpha \) and \(\beta \) with \(\alpha ,\,\beta >-1\). For Gegenbauer expansion, it is related to the parameter \(\lambda \) whatever f with interior or boundary singularities. All of these asymptotic analysis are optimal. Moreover, under the optimal asymptotic analysis, it derives that the truncated spectral expansions with some specific parameters can achieve the optimal convergence rates, i.e., the same as the best polynomial approximation in the sense of absolute maximum error norm. Numerical examples illustrate the perfect coincidence with the estimates.

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Notes

  1. One particularly interesting question is the comparison of the decay rates of the Chebyshev and Legendre coefficients. A myth on this issue is the “Lanczos–Fox–Parker” proposition [5] that the Chebyshev coefficient \(c_n\) decays approximately \(\frac{\sqrt{n\pi }}{2}\) faster than the Legendre coefficient \(a_{n}(1/2)\) for large values of n ([22, Lanczos] and Fox Parker [16, p. 17]).

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Acknowledgements

The authors are grateful for the referees’ helpful suggestions and insightful comments, which helped improve the manuscript significantly. The authors would like to thank for Prof. Li-Lian Wang for many constructive comments that helped to improve the presentation of this paper.

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Authors and Affiliations

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

    Shuhuang Xiang

  2. School of Statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, Jiangsu, People’s Republic of China

    Guidong Liu

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  1. Shuhuang Xiang
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  2. Guidong Liu
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Correspondence to Guidong Liu.

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This work was supported partly by NSF of China (No. 11771454)

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Xiang, S., Liu, G. Optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities. Numer. Math. 145, 117–148 (2020). https://doi.org/10.1007/s00211-020-01113-3

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