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Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases

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Abstract

When one solves differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials Tn(x) with coefficients an to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, u(±1) = 0, popular choices include the “Chebyshev difference basis” ςn(x) ≡ Tn+2(x)−Tn(x) with coefficients here denoted by bn and the “quadratic factor basis” ϱn(x) ≡ (1 − x2)Tn(x) with coefficients cn. If u(x) is weakly singular at the boundary, then the coefficients an decrease proportionally to \({\cal O}\left( {A\left( n \right)/{n^\kappa }} \right)\) for some positive constant κ, where A(n) is a logarithm or a constant. We prove that the Chebyshev difference coefficients bn decrease more slowly by a factor of 1/n while the quadratic factor coefficients cn decrease more slowly still as \({\cal O}\left( {A\left( n \right)/{n^{\kappa - 2}}} \right)\). The error for the unconstrained Chebyshev series, truncated at degree n = N, is \({\cal O}\left( {\left| {A\left( N \right)} \right|/{N^\kappa }} \right)\) in the interior, but is worse by one power of N in narrow boundary layers near each of the endpoints. Despite having nearly identical error norms in interpolation, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic factor and difference basis sets is nearly uniformly oscillating over the entire interval in x. Meanwhile, for Chebyshev polynomials, the values of their derivatives at the endpoints are \({\cal O}\left( {{N^2}} \right)\), but only \({\cal O}\left( N \right)\) for the difference basis. Furthermore, we give the asymptotic coefficients and rigorous error estimates of the approximations in these three bases, solved by the least squares method. We also find an interesting fact that on the face of it, the aliasing error is regarded as a bad thing; actually, the error norm associated with the downward curving spectral coefficients decreases even faster than the error norm of infinite truncation. But the premise is under the same basis, and when involving different bases, it may not be established yet.

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References

  1. Baszenski G, Delvos F J. Error estimates for sine series expansions. Math Nachr, 1988, 139: 155–166

    Article  MathSciNet  MATH  Google Scholar 

  2. Bernstein S. Remarques sur l’inégalité de Wladimir Markoff. Commun Soc Math Kharkow, 1913, 14: 81–87

    Google Scholar 

  3. Boyd J P. Spectral and pseudospectral methods for eigenvalue and nonseparable boundary value problems. Monthly Weather Rev, 1978, 106: 1192–1203

    Article  Google Scholar 

  4. Boyd J P. Chebyshev domain truncation is inferior to Fourier domain truncation for solving problems on an infinite interval. J Sci Comput, 1988, 3: 109–120

    Article  MathSciNet  MATH  Google Scholar 

  5. Boyd J P. Chebyshev and Legendre spectral methods in algebraic manipulation languages. J Symbolic Comput, 1993, 16: 377–399

    Article  MathSciNet  MATH  Google Scholar 

  6. Boyd J P. Chebyshev and Fourier Spectral Methods, 2nd ed. Mineola-New York: Dover, 2001

    MATH  Google Scholar 

  7. Boyd J P. Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms. J Engrg Math, 2009, 63: 355–399

    Article  MathSciNet  MATH  Google Scholar 

  8. Boyd J P. Solving Transcendental Equations: The Chebyshev Polynomial Proxy and Other Numerical Rootfinders, Perturbation Series and Oracles. Philadelphia: SIAM, 2014

    Book  MATH  Google Scholar 

  9. Boyd J P, Moore D W. Summability methods for Hermite functions. Dyn Atmos Oceans, 1986, 10: 51–62

    Article  Google Scholar 

  10. Boyd J P, Petschek R. The relationships between Chebyshev, Legendre and Jacobi polynomials: The generic superiority of Chebyshev polynomials and three important exceptions. J Sci Comput, 2014, 59: 1–27

    Article  MathSciNet  MATH  Google Scholar 

  11. Elliott D. The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function. Math Comp, 1964, 18: 274–284

    Article  MathSciNet  Google Scholar 

  12. Elliott D, Tuan P D. Asymptotic estimates of Fourier coefficients. SIAM J Math Anal, 1974, 5: 1–10

    Article  MathSciNet  MATH  Google Scholar 

  13. Ellison A C, Julien K, Vasil G M. A gyroscopic polynomial basis in the sphere. J Comput Phys, 2022, 460: 111170

    Article  MathSciNet  MATH  Google Scholar 

  14. Fox L, Parker I B. Chebyshev Polynomials in Numerical Analysis. London: Oxford University Press, 1968

    MATH  Google Scholar 

  15. Heinrichs W. Stabilization techniques for spectral methods. J Sci Compt, 1991, 6: 1–19

    Article  MathSciNet  MATH  Google Scholar 

  16. Hesthaven J S, Gottlieb S, Gottlieb D. Spectral Methods for Time-Dependent Problems. Cambridge: Cambridge University Press, 2007

    Book  MATH  Google Scholar 

  17. Karageorghis A. On the equivalence between basis recombination and boundary bordering formulations for spectral collocation methods in rectangular domains. Math Comput Simulation, 1993, 35: 113–123

    Article  MathSciNet  MATH  Google Scholar 

  18. Kzaz M. Asymptotic expansion of Fourier coefficients associated to functions with low continuity. J Comput Appl Math, 2000, 114: 217–230

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu W J, Wang L L, Li H Y. Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces. Math Comp, 2019, 88: 2857–2895

    Article  MathSciNet  MATH  Google Scholar 

  20. Mason J C, Handscomb D C. Chebyshev Polynomials, 1st ed. Boca Raton-London-New York: Chapman and Hall/CRC, 2002

    Book  MATH  Google Scholar 

  21. Morse P M, Feshbach H. Methods of Theoretical Physics, 1st ed. New York: McGraw-Hill, 1953

    MATH  Google Scholar 

  22. Olver F, Lozier D, Boisvert R, et al. The NIST Handbook of Mathematical Functions. New York: Cambridge University Press, 2010

    MATH  Google Scholar 

  23. Pearce C J. Transformation methods in the analysis of series for critical properties. Adv Phys, 1978, 27: 89–145

    Article  Google Scholar 

  24. Riess R D, Johnson L W. Error estimates for Clenshaw-Curtis quadrature. Numer Math, 1972, 18: 345–353

    Article  MathSciNet  MATH  Google Scholar 

  25. Snyder M A. Chebyshev Methods in Numerical Approximation. Englewood Cliffs: Prentice-Hall, 1966

    MATH  Google Scholar 

  26. Trefethen L N. Is Gauss quadrature better than Clenshaw-Curtis? SIAM Rev, 2008, 50: 67–87

    Article  MathSciNet  MATH  Google Scholar 

  27. Trefethen L N. Approximation Theory and Approximation Practice, Extended Edition. Philadelphia: SIAM, 2019

    Book  MATH  Google Scholar 

  28. Tuan P D, Elliott D. Coefficients in series expansions for certain classes of functions. Math Comp, 1972, 26: 213–232

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang H Y. On the optimal estimates and comparison of Gegenbauer expansion coefficients. SIAM J Numer Anal, 2016, 54: 1557–1581

    Article  MathSciNet  MATH  Google Scholar 

  30. Wang H Y. On the convergence rate of Clenshaw-Curtis quadrature for integrals with algebraic endpoint singularities. J Comput Appl Math, 2018, 333: 87–98

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang H Y. Are best approximations really better than Chebyshev? arXiv:2106.03456, 2021

  32. Whitley R. Bernstein’s asymptotic best bound for the kth derivative of a polynomial. J Math Anal Appl, 1985, 105: 502–513

    Article  MathSciNet  MATH  Google Scholar 

  33. Xiang S H. Convergence rates of spectral orthogonal projection approximation for functions of algebraic and logarithmatic regularities. SIAM J Numer Anal, 2021, 59: 1374–1398

    Article  MathSciNet  MATH  Google Scholar 

  34. Xiang S H, Bornemann F. On the convergence rates of Gauss and Clenshaw-Curtis quadrature for functions of limited regularity. SIAM J Numer Anal, 2012, 50: 2581–2587

    Article  MathSciNet  MATH  Google Scholar 

  35. Xiang S H, Liu G D. Optimal decay rates on the asymptotics of orthogonal polynomial expansions for functions of limited regularities. Numer Math, 2020, 145: 117–148

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhang X L. The mysteries of the best approximation and Chebyshev expansion for the function with logarithmic regularities. arXiv:2108.03836, 2021

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Acknowledgements

This work was supported by National Science Foundation of USA (Grant No. DMS-1521158), National Natural Science Foundation of China (Grant No. 12101229), the Hunan Provincial Natural Science Foundation of China (Grant No. 2021JJ40331) and the Chinese Scholarship Council (Grant Nos. 201606060017 and 202106720024). We thank the four referees, whose comments greatly improved the paper.

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Correspondence to Xiaolong Zhang.

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Zhang, X., Boyd, J.P. Asymptotic coefficients and errors for Chebyshev polynomial approximations with weak endpoint singularities: Effects of different bases. Sci. China Math. 66, 191–220 (2023). https://doi.org/10.1007/s11425-021-1974-x

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