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.
@article{bwmeta1.element.doi-10_2478_s11533-006-0016-7,
author = {Anry Nersessian and Arnak Poghosyan},
title = {Accelerating the convergence of trigonometric series},
journal = {Open Mathematics},
volume = {4},
year = {2006},
pages = {435-448},
zbl = {1126.65126},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0016-7}
}
Anry Nersessian; Arnak Poghosyan. Accelerating the convergence of trigonometric series. Open Mathematics, Tome 4 (2006) pp. 435-448. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-006-0016-7/
[1] G.A. Baker and P. Graves-Morris: Pade Approximants. Encyclopedia of mathematics and its applications, 2nd ed., Cambridge Univ. Press, Cambridge, 1996.
[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 | Zbl 0852.41016
[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 | Zbl 0666.65067
[4] E.W. Cheney: Introduction to Approximation Theory, McGraw-Hill, New York, 1996.
[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 | Zbl 0790.65014
[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 | Zbl 0830.65144
[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 | Zbl 0895.65067
[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] 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 | Zbl 0973.65133
[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 | Zbl 0844.42004
[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. | Zbl 0554.76058
[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] 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 | Zbl 0561.76076
[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] 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] 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 | Zbl 0852.42018
[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. | Zbl 0852.42017
[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] A. Krylov: On an approximate calculations, Lectures delivered in 1906 (in Russian), St Peterburg, Tipolitography of Birkenfeld, 1907.
[20] C. Lanczos: “Evaluation of noisy data”, J. Soc. Indust. Appl. Math., Ser. B Numer. Anal., Vol. 1, (1964), pp. 76–85. | Zbl 0142.12504
[21] C. Lanczos: Discourse on Fourier Series, Oliver and Boyd, Edinburgh, 1966.
[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] 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 | Zbl 0271.41006
[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] 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] 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] A. Nersessian and A. Poghosyan: “Asymptotic estimates for a nonlinear acceleration method of Fourier series”, Reports of NAS RA (in Russian), to be published. | Zbl 1114.41008
[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. | Zbl 1073.65571
[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. | Zbl 1098.41010
[30] A. Nersessian and A. Poghosyan: “On a rational linear approximation Fourier Series for smooth functions”, J. Sci. Comput., to be published. | Zbl 1114.41008
[31] S. Wolfram: The MATHEMATICA book, 4th ed., Wolfram Media, Cambridge University Press, 1999. | Zbl 0924.65002
[32] A. Zygmund: Trigonometric Series, Vol. 1,2, Cambridge Univ. Press, Cambridge, 1959. | Zbl 0085.05601