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Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces


Authors: Wenjie Liu, Li-Lian Wang and Huiyuan Li
Journal: Math. Comp. 88 (2019), 2857-2895
MSC (2010): Primary 41A10, 41A25, 41A50, 65N35, 65M60
DOI: https://doi.org/10.1090/mcom/3456
Published electronically: May 23, 2019
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Abstract: In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville fractional integrals/derivatives for optimal error estimates of Chebyshev polynomial approximations to functions with limited regularity. It naturally arises from exact representations of Chebyshev expansion coefficients. Here, the essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions), and (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties. Under this framework, we are able to estimate the optimal decay rate of Chebyshev expansion coefficients for a large class of functions with interior and endpoint singularities, which are deemed suboptimal or complicated to characterise in existing literature. Then we can derive optimal error estimates for spectral expansions and the related Chebyshev interpolation and quadrature measured in various norms, and also improve available results in usual Sobolev spaces with integer regularity exponentials in several senses. As a byproduct, this study results in some analytically perspicuous formulas particularly on GGF-Fs, which are potentially useful in spectral algorithms. The idea and analysis techniques can be extended to general Jacobi polynomial approximations.


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Additional Information

Wenjie Liu
Affiliation: Department of Mathematics, Harbin Institute of Technology, 150001, People’s Republic of China; and Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
Email: liuwenjie@hit.edu.cn

Li-Lian Wang
Affiliation: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371, Singapore
Email: lilian@ntu.edu.sg

Huiyuan Li
Affiliation: State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: huiyuan@iscas.ac.cn

DOI: https://doi.org/10.1090/mcom/3456
Keywords: Approximation by Chebyshev polynomials, fractional integrals/derivatives, fractional Sobolev-type spaces, singular functions, optimal estimates
Received by editor(s): October 24, 2017
Received by editor(s) in revised form: December 22, 2018
Published electronically: May 23, 2019
Additional Notes: The research of the first author was supported in part by a project funded by the China Postdoctoral Science Foundation (No. 2017M620113), the National Natural Science Foundation of China (Nos. 11801120, 71773024 and 11771107), the Fundamental Research Funds for the Central Universities (Grant No.HIT.NSRIF.2019058), and the Natural Science Foundation of Heilongjiang Province of China (No. G2018006)
The research of the second author was supported in part by Singapore MOE AcRF Tier 2 Grants (MOE2017-T2-2-144 and MOE2018-T2-1-059)
The research of the third author was supported in part by the National Natural Science Foundation of China (No. 11871455), the Strategic Priortity Research Program of Chinese Academy of Sciences (No. XDC01030200), and the National Key R&D Program of China (No. 2018YFB0204404).
Article copyright: © Copyright 2019 American Mathematical Society