Error bounds for approximation in Chebyshev points

Numerische Mathematik

Volumn 116, Issue 3, 2010, Pages 463-491

  Xiang, Shuhuang ^a   Chen, Xiaojun ^b   Wang, Haiyong ^a  

^a CENTRAL SOUTH UNIVERSITY   (China)

^b HONG KONG POLYTECHNIC UNIVERSITY   (Hong Kong)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77956177876     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-010-0309-4     Document Type: Article

References (32)

1. Adam G. H., Nobile A.: Product integration rules at Clenshaw-Curtis and related points: a robust implementation. IMA J. Numer. Math. 11, 271-296 (1991).
2. Abramowitz, M., Stegun, I. A.: Handbook of Mathematical Functions. National Bureau of Standards, Washington, D. C. (1964).
3. Bernstein S. N.: Sur l'ordre de la meilleure approximation des fonctions continues par les polyn?mes de degré donné. Mem. Cl. Sci. Acad. Roy. Belg. 4, 1-103 (1912).
4. Berrut J. P., Trefethen L. N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501-517 (2004).
5. Boyd J. P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2000).
6. Clenshaw C. W., Curtis A. R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197-205 (1960).
7. Dahlquist G., Björck A.: Numerical Methods in Scientific Computing. SIAM, Philadelphia (2007).
8. Davis P. J., Rabinowitz P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984).
9. Deaño A., Huybrechs D.: Complex Gaussian quadrature of oscillatory integrals. Numer. Math. 112, 197-219 (2009).
10. Evans G. A.: Practical Numerical Integration. Wiley, Chichester (1993).
11. Evans G. A., Webster J. R.: A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112, 55-69 (1999).
12. Glaser A., Liu X., Rokhlin V.: A fast algorithm for the calculation of the roots of special functions. SIAM J. Sci. Comput. 29, 1420-1438 (2007).
13. Gradshteyn I. S., Ryzhik I. M.: Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego (2000).
14. O'Hara H., Smith F. J.: Error estimation in the Clenshaw-Curtis quadrature formula. Comput. J. 11, 213-219 (1968).
15. r) on [0, 1]. J. Comput. Appl. Math. 223, 399-408 (2009).
16. Iserles A., Nørsett S. P.: Efficient quadrature of highly-oscillatory integrals using derivatives. Proc. R. Soc. A 461, 1383-1399 (2005).
17. Kussmaul R.: Clenshaw-Curtis quadrature with a weighting function. Computing 9, 159-164 (1972).
18. Littlewood R. K., Zakian V.: Numerical evaluation of Fourier integrals. J. Inst. Math. Appl. 18, 331-339 (1976).
19. Mason J. C., Handscomb D. C.: Chebyshev Polynomials. CRC Press, New York (2003).
20. Paterson T. N. L.: On high precision methods for the evaluation of Fourier integrals with finite and infinite limits. Numer. Math. 27, 41-52 (1976).
21. Piessens R., Branders M.: Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23, 370-381 (1983).
22. Piessens R., Poleunis F.: A numerical method for the integration of oscillatory functions. BIT 11, 317-327 (1971).
23. Powell M. J. D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981).
24. Sloan I. H.: On the numerical evaluation of singular integrals. BIT 18, 91-102 (1978).
25. Sloan I. H., Smith W. E.: Product-integration with the Clenshaw-Curtis and related points. Numer. Math. 30, 415-428 (1978).
26. Sloan I. H., Smith W. E.: Product integration with the Clenshaw-Curtis points: implementation and error estimates. Numer. Math. 34, 387-401 (1980).
27. Stein E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993).
28. Szegö G.: Orthogonal Polynomial. American Mathematical Society, Providence, Rhode Island (1939).
29. Trefethen L. N.: Spectral Methods in MATLAB. SIAM, Philadelphia (2000).
30. Trefethen L. N.: Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50, 67-87 (2008).
31. Waldvogel J.: Fast construction of the Fejér and Clenshaw-Curtis quadrature rules. BIT 46, 195-202 (2006).
32. iωg(x)dx. Numer. Math. 105, 633-658 (2007).