Abstract
In this paper we present a new perspective on error analysis for Legendre approximations of differentiable functions. We start by introducing a sequence of Legendre–Gauss–Lobatto polynomials and prove their theoretical properties, including an explicit and optimal upper bound. We then apply these properties to derive a new explicit bound for the Legendre coefficients of differentiable functions. Building on this, we establish an explicit and optimal error bound for Legendre approximations in the \(L^2\) norm and an explicit and optimal error bound for Legendre approximations in the \(L^{\infty }\) norm under the condition that their maximum error is attained in the interior of the interval. Illustrative examples are provided to demonstrate the sharpness of our new results.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant 11671160. The author wishes to thank the editor and two anonymous referees for their valuable comments on the manuscript.
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Communicated by Arieh Iserles.
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Wang, H. New Error Bounds for Legendre Approximations of Differentiable Functions. J Fourier Anal Appl 29, 42 (2023). https://doi.org/10.1007/s00041-023-10024-4
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DOI: https://doi.org/10.1007/s00041-023-10024-4
Keywords
- Legendre approximations
- Differentiable functions
- Legendre coefficients
- Legendre–Gauss–Lobatto functions
- Optimal convergence rates