On some quasi-periodic approximations
DOI:
https://doi.org/10.52737/18291163-2020.12.10-1-27Keywords:
Fourier series, trigonometric interpolation, convergence acceleration, quasi-periodic approximation, quasi-periodic interpolationAbstract
Trigonometric approximation or interpolation of a non-smooth function on a finite interval has poor convergence properties. This is especially true for discontinuous functions. The case of infinitely differentiable but non-periodic functions with discontinuous periodic extensions onto the real axis has attracted interest from many researchers. In a series of works, we discussed an approach based on quasi-periodic trigonometric basis functions whose periods are slightly bigger than the length of the approximation interval. We proved validness of the approach for trigonometric interpolations. In this paper, we apply those ideas to classical Fourier expansions.
References
A. Zygmund. Trigonometric Series. Vol. 1,2. Cambridge Univ. Press, 1959.
A. J. Jerri. The Gibbs phenomenon in Fourier analysis, splines and wavelet approximations, volume 446 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 1998.
G. Helmberg. The Gibbs phenomenon for Fourier interpolation. J. Approx. Theory, 78(1):41-63, 1994.
W. B. Jones and G. Hardy. Accelerating convergence of trigonometric approximations. Math. Comp., 24:547-560, 1970.
G. Baszenski, F.-J. Delvos, and M. Tasche. A united approach to accelerating trigonometric expansions. Computers Math. Applic., 30(3-6):33-49, 1995.
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., 71(4):511-526, 1995.
A. Nersessian and A. Poghosyan. Accelerating the convergence of trigonometric series. Cent. Eur. J. Math., 4(3):435-448, 2006.
J. P. Boyd. Acceleration of algebraically-converging Fourier series when the coefficients have series in powers in 1=n. J. Comput. Phys., 228(5):1404-1411, 2009.
A. Poghosyan. Asymptotic behavior of the Krylov-Lanczos interpolation. Anal. Appl. (Singap.), 7(2):199-211, 2009.
A. Poghosyan. Asymptotic behavior of the Eckhoff method for convergence acceleration of trigonometric interpolation. Anal. Theory Appl., 26(3):236-260, 2010.
A. Poghosyan. On an auto-correction phenomenon of the Krylov-Gottlieb-Eckhoff method. IMA J. Numer. Anal., 31(2):512-527, 2011.
D. Batenkov and Y. Yomdin. Algebraic Fourier reconstruction of piecewise smooth functions. Math. Comp., 81(277):277-318, 2012.
A. Pogosyan. On the convergence of rational-trigonometric-polynomial approximations realized by roots of Laguerre polynomials. Izv. Nats. Akad. Nauk Armenii Mat., 48(6):82-91, 2013.
D. Batenkov. Complete algebraic reconstruction of piecewise-smooth functions from Fourier data. Math. Comp., 84(295):2329-2350, 2015.
A. Nersessian. On an over-convergence phenomenon for Fourier series. Basic approach. Armen. J. Math., 10:Paper No. 9, 22, 2018.
A. Nersessian. A correction to the article "On an over-convergence
phenomenon for Fourier series. Basic approach" [ MR3870875]. Armen. J. Math., 11:Paper No. 1, 2, 2019.
A. Nersessian. Convergence acceleration of Fourier series revisited. Armen. J. Math., 3(4):152-161, 2010.
A. Nersessian and N. Oganesyan. Quasiperiodic interpolation. Reports of NAS RA, 101(2):115-121, 2001.
L. Poghosyan. On L2-convergence of the quasi-periodic interpolation. Dokl. Nats. Akad. Nauk Armen., 113(3):240-247, 2013.
L. Poghosyan and A. Poghosyan. Asymptotic estimates for the quasiperiodic interpolations. Armen. J. Math., 5(1):34-57, 2013.
A. Poghosyan and L. Poghosyan. On a pointwise convergence of quasiperiodic-
rational trigonometric interpolation. Int. J. Anal., pages Art.
ID 249513, 10, 2014.
L. Pogosyan and A. Pogosyan. On the pointwise convergence of a quasiperiodic trigonometric interpolation. Izv. Nats. Akad. Nauk Armenii Mat., 49(3):68-80, 2014.
J. N. Lyness. Computational techniques based on the Lanczos representation. Math. Comp., 28:81-123, 1974.
A. Barkhudaryan, R. Barkhudaryan, and A. Poghosyan. Asymptotic behavior of Eckhoff's method for Fourier series convergence acceleration. Anal. Theory Appl., 23(3):228-242, 2007.
A. Bjork and V. Pereyra. Solution of Vandermonde systems of equations. Math. Comp., 24:893-904, 1970.
N. J. Higham. Error analysis of the Björck-Pereyra algorithms for solving Vandermonde systems. Numer. Math., 50(5):613-632, 1987.
L. Poghosyan. Convergence acceleration of quasi-periodic and quasiperiodic-rational interpolations by polynomial corrections. Armen. J. Math., 5(2):123-138, 2013.
B. Adcock. Convergence acceleration of modified Fourier series in one or more dimensions. Math. Comp., 80(273):225-261, 2011.
B. Adcock. Modified fourier expansions: theory, construction and applications. PhD Thesis, University of Cambridge, 2010.
T. Bakaryan. On a convergence of the modified Fourier-Pade approximations. Armen. J. Math., 8(2):120-144, 2016.
A. Pogosyan and T. Bakaryan. On interpolation with respect to a modified trigonometric system. Izv. Nats. Akad. Nauk Armenii Mat., 53(3):72-83, 2018.
A. Poghosyan and T. Bakaryan. Optimal rational approximations by the modified Fourier basis. Abstr. Appl. Anal., pages Art. ID 1705409, 21, 2018.
Downloads
Published
Versions
- 2022-09-02 (2)
- 2020-10-30 (1)
Issue
Section
License
Copyright (c) 2020 Armenian Journal of Mathematics
This work is licensed under a Creative Commons Attribution 4.0 International License.
