On the convergence rates of Legendre approximation

Wang, Haiyong; Xiang, Shuhuang

doi:10.1090/S0025-5718-2011-02549-4

On the convergence rates of Legendre approximation

Authors: Haiyong Wang and Shuhuang Xiang
Journal: Math. Comp. 81 (2012), 861-877
MSC (2010): Primary 65D05, 65D99, 41A25
DOI: https://doi.org/10.1090/S0025-5718-2011-02549-4
Published electronically: October 18, 2011
MathSciNet review: 2869040
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of the rate of convergence of Legendre approximation is considered. We first establish the decay rates of the coefficients in the Legendre series expansion and then derive error bounds of the truncated Legendre series in the uniform norm. In addition, we consider Legendre approximation with interpolation. In particular, we are interested in the barycentric Lagrange formula at the Gauss-Legendre points. Explicit barycentric weights, in terms of Gauss-Legendre points and corresponding quadrature weights, are presented that allow a fast evaluation of the Legendre interpolation formula. Error estimates for Legendre interpolation polynomials are also given.

1. Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642
2. Forman S. Acton, Numerical methods that work, Mathematical Association of America, Washington, DC, 1990. Corrected reprint of the 1970 edition. MR 1074173
3. Bradley K. Alpert and Vladimir Rokhlin, A fast algorithm for the evaluation of Legendre expansions, SIAM J. Sci. Statist. Comput. 12 (1991), no. 1, 158–179. MR 1078802, https://doi.org/10.1137/0912009
4. Jean-Paul Berrut and Lloyd N. Trefethen, Barycentric Lagrange interpolation, SIAM Rev. 46 (2004), no. 3, 501–517. MR 2115059, https://doi.org/10.1137/S0036144502417715
5. John P. Boyd, Chebyshev and Fourier spectral methods, 2nd ed., Dover Publications, Inc., Mineola, NY, 2001. MR 1874071
6. H. Brunner, A. Iserles and S. P. Nørsett, The computation of the spectra of highly oscillatory Fredholm operators, J. Integral Equations Appl., to appear.
7. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral methods, Scientific Computation, Springer-Verlag, Berlin, 2006. Fundamentals in single domains. MR 2223552
8. E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
9. C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2 (1960), 197–205. MR 117885, https://doi.org/10.1007/BF01386223
10. Germund Dahlquist and Åke Björck, Numerical methods in scientific computing. Vol. I, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. MR 2412832
11. Philip J. Davis, Interpolation and approximation, Dover Publications, Inc., New York, 1975. Republication, with minor corrections, of the 1963 original, with a new preface and bibliography. MR 0380189
12. Philip J. Davis and Philip Rabinowitz, Methods of numerical integration, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR 760629
13. Bengt Fornberg, A practical guide to pseudospectral methods, Cambridge Monographs on Applied and Computational Mathematics, vol. 1, Cambridge University Press, Cambridge, 1996. MR 1386891
14. L. Fox and I. B. Parker, Chebyshev polynomials in numerical analysis, Oxford University Press, London-New York-Toronto, Ont., 1968. MR 0228149
15. Daniele Funaro, Spectral elements for transport-dominated equations, Lecture Notes in Computational Science and Engineering, vol. 1, Springer-Verlag, Berlin, 1997. MR 1449871
16. Walter Gautschi, Numerical analysis, Birkhäuser Boston, Inc., Boston, MA, 1997. An introduction. MR 1454125
17. Andreas Glaser, Xiangtao Liu, and Vladimir Rokhlin, A fast algorithm for the calculation of the roots of special functions, SIAM J. Sci. Comput. 29 (2007), no. 4, 1420–1438. MR 2341794, https://doi.org/10.1137/06067016X
18. Gene H. Golub and John H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), 221-230; addendum, ibid. 23 (1969), no. 106, loose microfiche suppl, A1–A10. MR 0245201, https://doi.org/10.1090/S0025-5718-69-99647-1
19. David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. MR 0520152
20. Ben-Yu Guo, Spectral methods and their applications, World Scientific Publishing Co., Inc., River Edge, NJ, 1998. MR 1641586
21. Jan S. Hesthaven, Sigal Gottlieb, and David Gottlieb, Spectral methods for time-dependent problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21, Cambridge University Press, Cambridge, 2007. MR 2333926
22. Nicholas J. Higham, The numerical stability of barycentric Lagrange interpolation, IMA J. Numer. Anal. 24 (2004), no. 4, 547–556. MR 2094569, https://doi.org/10.1093/imanum/24.4.547
23. A. Iserles, A fast and simple algorithm for the computation of Legendre coefficients, Numer. Math., 217 (2011), 529-553.
24. Jae-Hun Jung and Bernie D. Shizgal, On the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon, J. Comput. Phys. 224 (2007), no. 2, 477–488. MR 2330280, https://doi.org/10.1016/j.jcp.2007.01.018
25. J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1937591
26. R. Piessens, Computation of Legendre series coefficients, Algorithm 473, Comm. ACM., 17 (1974), 25.
27. Earl D. Rainville, Special functions, The Macmillan Co., New York, 1960. MR 0107725
28. H. E. Salzer, Lagrangian interpolation at the Chebyshev points 𝑋_{𝑛,𝜈}≡𝑐𝑜𝑠(𝜈𝜋/𝑛), 𝜈=0(1)𝑛; some unnoted advantages, Comput. J. 15 (1972), 156–159. MR 315865, https://doi.org/10.1093/comjnl/15.2.156
29. Jie Shen, Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput. 15 (1994), no. 6, 1489–1505. MR 1298626, https://doi.org/10.1137/0915089
30. Jie Shen, Efficient spectral-Galerkin method. II. Direct solvers of second- and fourth-order equations using Chebyshev polynomials, SIAM J. Sci. Comput. 16 (1995), no. 1, 74–87. MR 1311679, https://doi.org/10.1137/0916006
31. Jie Shen and Tao Tang, Spectral and high-order methods with applications, Mathematics Monograph Series, vol. 3, Science Press Beijing, Beijing, 2006. MR 2723481
32. P. K. Suetin, Representation of continuous and differentiable functions by Fourier series of Legendre polynomials, Soviet Math. Dokl., 5 (1964), 1408-1410.
33. Endre Süli and David F. Mayers, An introduction to numerical analysis, Cambridge University Press, Cambridge, 2003. MR 2006500
34. Gabor Szegö, Orthogonal Polynomials, American Mathematical Society, New York, 1939. American Mathematical Society Colloquium Publications, v. 23. MR 0000077
35. Lloyd N. Trefethen, Is Gauss quadrature better than Clenshaw-Curtis?, SIAM Rev. 50 (2008), no. 1, 67–87. MR 2403058, https://doi.org/10.1137/060659831
36. Lloyd N. Trefethen, Spectral methods in MATLAB, Software, Environments, and Tools, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. MR 1776072
37. L. N. Trefethen, N. Hale, R. B. Platte, T. A. Driscoll and R. Pachón, Chebfun Version 3, http://www.maths.ox.ac.uk/chebfun/, University of Oxford, 2009.
38. P. Vértesi, Lagrange interpolation for continuous functions of bounded variation, Acta Math. Acad. Sci. Hungar. 35 (1980), no. 1-2, 23–31. MR 588875, https://doi.org/10.1007/BF01896819
39. H. Wang and D. Huybrechs, Explicit barycentric weights for polynomial interpolation in the roots of extrema of classical orthogonal polynomials, in preparation.
40. Shuhuang Xiang, Xiaojun Chen, and Haiyong Wang, Error bounds for approximation in Chebyshev points, Numer. Math. 116 (2010), no. 3, 463–491. MR 2684294, https://doi.org/10.1007/s00211-010-0309-4