×

Accelerating the convergence of trigonometric series. (English) Zbl 1126.65126

Summary: 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.

MSC:

65T40 Numerical methods for trigonometric approximation and interpolation
42A15 Trigonometric interpolation
42A20 Convergence and absolute convergence of Fourier and trigonometric series

Software:

Mathematica
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G.A. Baker and P. Graves-Morris: Pade Approximants. Encyclopedia of mathematics and its applications, 2nd ed., Cambridge Univ. Press, Cambridge, 1996.; · Zbl 0923.41001
[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; · Zbl 0781.42022
[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.; · Zbl 0163.07601
[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.; · Zbl 0457.65068
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.