Browse > Article
http://dx.doi.org/10.14317/jami.2013.263

FOURIER SERIES ACCELERATION AND HARDY-LITTLEWOOD SERIES  

Ciszewski, Regina (Department of Mathematics and Statistics Mount Holyoke College South Hadley)
Gregory, Jason (Department of Mathematics and Statistics Murray State University Murray)
Moore, Charles N. (Department of Mathematics, Kansas State University)
West, Jasmine (Department of Mathematics, Spelman College Atlanta)
Publication Information
Journal of applied mathematics & informatics / v.31, no.1_2, 2013 , pp. 263-276 More about this Journal
Abstract
We discuss the effects of the ${\delta}^2$ and Lubkin acceleration methods on the partial sums of Fourier Series. We construct continuous, even H$\ddot{o}$lder continuous functions, for which these acceleration methods fail to give convergence. The constructed functions include some interesting trigonometric series whose properties were investigated by Hardy and Littlewood.
Keywords
Sequence acceleration; Fourier series;
Citations & Related Records
연도 인용수 순위
  • Reference
1 D. Shanks, Non-linear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955), 1-42.   DOI
2 A. Sidi,Practical Extrapolation Methods, Cambridge Monographs on Applied and Computational Mathematics 10, Cambridge University Press, Cambridge, 2003.
3 P, Wynn, On a Device for Computing the em(Sn) Transformation, Math. Tables Aids Comput. 10 (1956), 91-96.   DOI
4 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 Aeorspace Research, United States Air Force, 1967), 339-379.
5 A. Zygmund,Trigonometric Series, Second Edition, Cambridge University Press, Cambridge, 1959.
6 E. Abebe, J. Graber, and C. N. Moore, Fourier series and the ${\delta}^2$ process, J. Comput. Appl. Math. 224, no.1 (2009), 146-151.   DOI   ScienceOn
7 B. Beckermann, A. Matos, and F. Wielonsky, Reduction of the Gibbs phenomenon for smooth functions with jumps by the ${\epsilon}$-algorithm, J. Comput. Appl. Math. 219 (2008), 329-349.   DOI   ScienceOn
8 J. Boggess, E. Bunch, and C. N. Moore, Fourier series and the Lubkin W-transform, Numer. Algorithms 47 (2008), 133-142.   DOI
9 C. Brezinski, Extrapolation algorithms for filtering series of functions, and treating the Gibbs phenomenon, Numer. Algorithms 36 (2004), 309-329.   DOI
10 C. Brezinski and M. Redivo-Zaglia, Extrapolation Methods. Theory and Practice. North- Holland, Amsterdam, 1991.
11 G. H. Hardy and J. E. Littlewood, Some problems of Diophantine approximation: A remarkable trigonometric series, Proc. Natl. Acad. Sci. USA 2 (1916), 583-586.   DOI   ScienceOn
12 S. Lubkin, A method for summing infinite series, J. Res. Natl. Bur. Stand. 48 (1952), 228-254.   DOI