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How much faster does the best polynomial approximation converge than Legendre projection?

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Abstract

We compare the convergence behavior of best polynomial approximations and Legendre and Chebyshev projections and derive optimal rates of convergence of Legendre projections for analytic and differentiable functions in the maximum norm. For analytic functions, we show that the best polynomial approximation of degree n is better than the Legendre projection of the same degree by a factor of \(n^{1/2}\). For differentiable functions such as piecewise analytic functions and functions of fractional smoothness, however, we show that the best approximation is better than the Legendre projection by only some constant factors. Our results provide some new insights into the approximation power of Legendre projections.

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Acknowledgements

The author would like to thank two anonymous referees for their careful reading of the manuscript and helpful comments which have improved this paper.

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  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Haiyong Wang

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Haiyong Wang

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Correspondence to Haiyong Wang.

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This work was supported by National Natural Science Foundation of China under Grant number 11671160.

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Wang, H. How much faster does the best polynomial approximation converge than Legendre projection?. Numer. Math. 147, 481–503 (2021). https://doi.org/10.1007/s00211-021-01173-z

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