Accelerating the convergence of trigonometric series

Anry Nersessian; Arnak Poghosyan

Open Mathematics (2006)

  • Volume: 4, Issue: 3, page 435-448
  • ISSN: 2391-5455

Abstract

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A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.

How to cite

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Anry Nersessian, and Arnak Poghosyan. "Accelerating the convergence of trigonometric series." Open Mathematics 4.3 (2006): 435-448. <http://eudml.org/doc/269008>.

@article{AnryNersessian2006,
abstract = {A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.},
author = {Anry Nersessian, Arnak Poghosyan},
journal = {Open Mathematics},
keywords = {65B99; 42A10; 42A15; 41A21},
language = {eng},
number = {3},
pages = {435-448},
title = {Accelerating the convergence of trigonometric series},
url = {http://eudml.org/doc/269008},
volume = {4},
year = {2006},
}

TY - JOUR
AU - Anry Nersessian
AU - Arnak Poghosyan
TI - Accelerating the convergence of trigonometric series
JO - Open Mathematics
PY - 2006
VL - 4
IS - 3
SP - 435
EP - 448
AB - A nonlinear method of accelerating both the convergence of Fourier series and trigonometric interpolation is investigated. Asymptotic estimates of errors are derived for smooth functions. Numerical results are represented and discussed.
LA - eng
KW - 65B99; 42A10; 42A15; 41A21
UR - http://eudml.org/doc/269008
ER -

References

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  1. [1] G.A. Baker and P. Graves-Morris: Pade Approximants. Encyclopedia of mathematics and its applications, 2nd ed., Cambridge Univ. Press, Cambridge, 1996. 
  2. [2] G. Baszenski, F.-J. Delvos and M. Tasche: “A united approach to accelerating trigonometric expansions”, Comput. Math. Appl., Vol. 30(3–6), (1995), pp. 33–49. http://dx.doi.org/10.1016/0898-1221(95)00084-4 Zbl0852.41016
  3. [3] W. Cai, D. Gottlieb and C.W. Shu: “Essentially non oscillatory spectral Fourier methods for shock wave calculations”, Math. Comp., Vol. 52, (1989), pp. 389–410. http://dx.doi.org/10.2307/2008473 Zbl0666.65067
  4. [4] E.W. Cheney: Introduction to Approximation Theory, McGraw-Hill, New York, 1996. 
  5. [5] K.S. Eckhoff: “Accurate and efficient reconstruction of discontinuous functions from truncated series expansions”, Math. Comp., Vol. 61, (1993), pp. 745–763. http://dx.doi.org/10.2307/2153251 Zbl0790.65014
  6. [6] K.S. Eckhoff: “Accurate reconstructions of functions of finite regularity from truncated Fourier series expansions”, Math. Comp., Vol. 64, (1995), pp. 671–690. http://dx.doi.org/10.2307/2153445 Zbl0830.65144
  7. [7] K.S. Eckhoff: “On a high order numerical method for functions with singularities”, Math. Comp., Vol. 67, (1998), pp. 1063–1087. http://dx.doi.org/10.1090/S0025-5718-98-00949-1 Zbl0895.65067
  8. [8] C.E. Wasberg: On the numerical approximation of derivatives by a modified Fourier collocation method, Thesis (PhD), Department of Mathematics, University of Bergen, Norway, 1996. 
  9. [9] T.A. Driscoll and B. Fornberg: “A Pade-based algorithm for overcoming the Gibbs phenomenon”, Numerical Algorithms, Vol. 26, (2000), pp. 77–92. http://dx.doi.org/10.1023/A:1016648530648 Zbl0973.65133
  10. [10] J. Geer: “Rational trigonometric approximations using Fourier series partial sums”, J. Sci. Computing, Vol. 10(3), (1995), pp. 325–356. http://dx.doi.org/10.1007/BF02091779 Zbl0844.42004
  11. [11] D. Gottlieb: “Spectral methods for compressible flow problems”, In: Soubbaramayer and J.P. Boujot (Eds.): Proc. 9th Internat. Conf. Numer. Methods Fluid Dynamics, Lecture Notes in Phys., Vol. 218, Saclay, France, Springer-Verlag, Berlin and New York, 1985, pp. 48–61. Zbl0554.76058
  12. [12] D. Gottlieb: “Issues in the application of high order schemes”, In: M.Y. Hussaini, A. Kumar and M.D. Salas (Eds): Proc. Workshop on Algorithmic Trends in Computational Fluid Dynamics (Hampton, Virginia, USA), Springer-Verlag, ICASE /NASA LaRC Series, 1991, pp. 195–218. 
  13. [13] D. Gottlieb, L. Lustman and S.A. Orszag: “Spectral calculations of one-dimensional inviscid compressible flows”, SIAM J. Sci. Statist. Comput., Vol. 2, (1981), pp. 296–310. http://dx.doi.org/10.1137/0902024 Zbl0561.76076
  14. [14] D. Gottlieb, C.W. Shu, A. Solomonoff and H. Vandevon: “On the Gibbs Phenomenon I: Recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function”, J. Comput. Appl. Math., Vol. 43, (1992), pp. 81–92. http://dx.doi.org/10.1016/0377-0427(92)90260-5 
  15. [15] D. Gottlieb and C.W. Shu: On the Gibbs Phenomenon III: Recovering Exponential Accuracy in a sub-interval from the spectral partial sum of a piecewise analytic function, ICASE report, 1993, pp. 93–82. 
  16. [16] D. Gottlieb and C.W. Shu: “On the Gibbs phenomena IV: Recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function”, Math. Comp., Vol. 64, (1995), pp. 1081–1096. http://dx.doi.org/10.2307/2153484 Zbl0852.42018
  17. [17] D. Gottlieb and C.W. Shu: “On the Gibbs Phenomenon V: Recovering Exponential Accuracy from collocation point values of a piecewise analytic function”, Numer. Math., Vol. 33, (1996), pp. 280–290. Zbl0852.42017
  18. [18] W.B. Jones and G. Hardy: “Accelerating Convergence of Trigonometric Approximations”, Math. Comp., Vol. 24, (1970), pp. 47–60. http://dx.doi.org/10.2307/2004830 
  19. [19] A. Krylov: On an approximate calculations, Lectures delivered in 1906 (in Russian), St Peterburg, Tipolitography of Birkenfeld, 1907. 
  20. [20] C. Lanczos: “Evaluation of noisy data”, J. Soc. Indust. Appl. Math., Ser. B Numer. Anal., Vol. 1, (1964), pp. 76–85. Zbl0142.12504
  21. [21] C. Lanczos: Discourse on Fourier Series, Oliver and Boyd, Edinburgh, 1966. 
  22. [22] P.D. Lax: “Accuracy and resolution in the computation of solutions of linear and nonlinear equations”, In: C. de Boor and G.H. Golub (Eds.): Recent Advances in Numerical Analysis, Proc. Symposium Univ of Wisconsin-Madison, Academic Press, New York, 1978, pp. 107–117. 
  23. [23] J.N. Lyness: “Computational Techniques Based on the Lanczos Representation”, Math. Comp., Vol. 28, (1974), pp. 81–123. http://dx.doi.org/10.2307/2005818 Zbl0271.41006
  24. [24] A. Nersessian: “Bernoulli type quasipolynomials and accelerating convergence of Fourier Series of piecewise smooth functions (in Russian)”, Reports of NAS RA, Vol. 104(4), (2004), pp. 186–191. 
  25. [25] A. Nersessian and A. Poghosyan: “Bernoulli method in multidimensional case”, Preprint No20 Ar-00, Deposited in ArmNIINTI 09.03.00, (2000), pp. 1–40 (in Russian). 
  26. [26] A. Nersessian and A. Poghosyan: “On a rational linear approximation on a finite interval”, Reports of NAS RA, Vol. 104(3), (2004), pp. 177–184 (in Russian). 
  27. [27] A. Nersessian and A. Poghosyan: “Asymptotic estimates for a nonlinear acceleration method of Fourier series”, Reports of NAS RA (in Russian), to be published. Zbl1114.41008
  28. [28] A. Nersessian and A. Poghosyan: “Asymptotic errors of accelerated two-dimensional trigonometric approximations”, In: G.A. Barsegian, H.G.W. Begehr, H.G. Ghazaryan and A. Nersessian (Eds.): Complex Analysis, Differential Equations and Related Topics, Yerevan, Armenia, September 17–21, 2002, “Gitutjun” Publishing House, Yerevan, Armenia, 2004, pp. 70–78. Zbl1073.65571
  29. [29] A. Poghosyan: “On a convergence of a rational trigonometric approximation”, In: G.A. Barsegian, H.G.W. Begehr, H.G. Ghazaryan and A. Nersessian (Eds.): Complex Analysis, Differential Equations and Related Topics, Yerevan, Armenia, September 17–21, 2002, “Gitutjun” Publishing House, Yerevan, Armenia, 2004, pp. 79–87. Zbl1098.41010
  30. [30] A. Nersessian and A. Poghosyan: “On a rational linear approximation Fourier Series for smooth functions”, J. Sci. Comput., to be published. Zbl1114.41008
  31. [31] S. Wolfram: The MATHEMATICA book, 4th ed., Wolfram Media, Cambridge University Press, 1999. Zbl0924.65002
  32. [32] A. Zygmund: Trigonometric Series, Vol. 1,2, Cambridge Univ. Press, Cambridge, 1959. Zbl0085.05601

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