Extrapolation algorithms for filtering series of functions, and treating the Gibbs phenomenon

Numerical Algorithms

Volumn 36, Issue 4, 2004, Pages 309-329

  Brezinski, C ^a  

^a UNIVERSITÉ LILLE 1   (France)

Author keywords

Acceleration of convergence; Fourier series; Gibbs phenomenon; Series of functions

Indexed keywords

EID: 26844452802     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11075-004-2843-6     Document Type: Article

References (32)

1. G.A. Baker, Jr. and P.R. Graves-Morris, Padé Approximants, 2nd ed. (Cambridge University Press, Cambridge, 1996).
2. C. Bernardi and Y. Maday, Approximations Spectrales de Problèmes aux Limites Elliptiques (Springer-Verlag, Paris, 1992).
3. C. Brezinski, Algorithmes d'Accélération de la Convergence. Étude Numérique (Technip, Paris, 1978).
4. C. Brezinski, Padé-Type Approximation and General Orthogonal Polynomials (Birkhäuser, Basel, 1980).
5. C. Brezinski, How to accelerate continued fractions, in: Informatique et Calcul, eds. P. Chenin et al. (Masson, Paris, 1986) pp. 35-39.
6. C. Brezinski, Computational Aspects of Linear Control (Kluwer, Dordrecht, 2002).
7. C. Brezinski and M. Redivo Zaglia, Extrapolation Methods. Theory and Practice (North-Holland, Amsterdam, 1991).
8. A. Bultheel, Laurent Series and their Padé Approximations (Birkhäuser, Basel, 1987).
9. C.W. Clenshaw and K. Lord, Rational approximations from Chebyshev series, in: Studies in Numerical Analysis, ed. B.K.P. Scaife (Academic Press, London, 1974) pp. 95-113.
10. T.A. Driscoll and B. Fornberg, A Padé-based algorithm for overcoming the Gibbs phenomenon, Numer. Algorithms 26 (2001) 77-92.
11. L. Emmel, S.M. Kaber and Y. Maday, Padé-Jacobi filtering for spectral approximations of discontinuous solutions, Numer. Algorithms 33 (2003) 251-264.
12. S. Engelberg, Concentration factors, Preprint, Center for Scientific Computation and Mathematical Modeling, University of Maryland, College Park, USA (2003).
13. J. Fleischer, Analytic continuation of scattering amplitudes and Padé approximants, Nucl. Phys. B 37 (1972) 59-76 (erratum: ibid. B44 (1972) 641).
14. J. Fleischer, Nonlinear Padé approximants for Legendre series, J. Math. Phys. 14 (1973) 246-248.
15. A. Gelb and E. Tadmor, Detection of edges in spectral data, Appl. Comput. Harmonic Anal. 7 (1999) 101-135.
16. D. Gottlieb and S.A. Orszag, Numerical Analysis of Spectral Methods. Theory and Applications (SIAM, Philadelphia, 1977).
17. D. Gottlieb and C.W. Shu, The Gibbs phenomenon and its resolution, SIAM Rev. 39 (1997) 644-668.
18. W.B. Gragg, Laurent, Fourier, and Chebyshev-Padé tables, in: Padé and Rational Approximation. Theory and Applications, eds. E.B. Saff and R.S. Varga (Academic Press, New York, 1977) pp. 6172.
19. A. Gray and M. Pinsky, Gibbs phenomenon for Fourier-Bessel series, Exposition. Math. 11 (1993) 123-135.
20. C. Guilpin, J. Gacougnolle and Y. Simon, The ε-algorithm allows to detect εDirac delta functions, Appl. Numer. Math. 48 (2004) 27-40.
21. J.T. Holdeman, Jr., A method for the approximation of functions defined by formal series expansions in orthogonal polynomials, Math. Comput. 23 (1969) 275-288.
22. A.J. Jerri, The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations (Kluwer, Dordrecht, 1998).
23. S.M. Kaber and Y. Maday, Analysis of some Padé-Chebyshev approximants, submitted.
24. N.N. Lebedev, Special Functions and their Applications (Dover, New York, 1972).
25. A.C. Matos, Recursive computation of Padé-Legendre approximants and some acceleration properties, Numer. Math. 89 (2001) 535-560.
26. J.M.A. Matos and Z. da Rocha, Frobenius-Padé approximants for d-orthogonal series: Theory and applications, Appl. Numer. Math., to appear.
27. S. Paszkowski, Approximation uniforme des fonctions continues par des fonctions rationnelles, Zastos. Mat. 6 (1963) 441-458.
28. A. Peyerimhoff, Lectures on Summability, Lecture Notes in Math., Vol. 107 (Springer-Verlag, Berlin, 1969).
29. F. Stenger, Numerical Methods Based on Sine and Analytic Functions (Springer-Verlag, New York, 1993).
30. E.J. Weniger, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series, Comput. Phys. Rep. 10 (1989) 189-371.
31. J. Wimp, Sequence Transformations and their Applications (Academic Press, New York, 1981).
32. P. Wynn, Transformations to accelerate the convergence of Fourier series, in: Gertrude Blanch Anniversary Volume (Wright Patterson Air Force Base, Aerospace Research Laboratories, Office of Aerospace Research, United States Air Force, 1967) pp. 339-379.