Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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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)
  • Received : 2012.02.20
  • Accepted : 2012.10.15
  • Published : 2013.01.30
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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.

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References

  1. 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. https://doi.org/10.1016/j.cam.2008.04.020
  2. 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. https://doi.org/10.1016/j.cam.2007.11.011
  3. J. Boggess, E. Bunch, and C. N. Moore, Fourier series and the Lubkin W-transform, Numer. Algorithms 47 (2008), 133-142. https://doi.org/10.1007/s11075-007-9151-x
  4. C. Brezinski, Extrapolation algorithms for filtering series of functions, and treating the Gibbs phenomenon, Numer. Algorithms 36 (2004), 309-329. https://doi.org/10.1007/s11075-004-2843-6
  5. C. Brezinski and M. Redivo-Zaglia, Extrapolation Methods. Theory and Practice. North- Holland, Amsterdam, 1991.
  6. 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. https://doi.org/10.1073/pnas.2.10.583
  7. S. Lubkin, A method for summing infinite series, J. Res. Natl. Bur. Stand. 48 (1952), 228-254. https://doi.org/10.6028/jres.048.032
  8. D. Shanks, Non-linear transformations of divergent and slowly convergent sequences, J. Math. Phys. 34 (1955), 1-42. https://doi.org/10.1002/sapm19553411
  9. A. Sidi,Practical Extrapolation Methods, Cambridge Monographs on Applied and Computational Mathematics 10, Cambridge University Press, Cambridge, 2003.
  10. P, Wynn, On a Device for Computing the em(Sn) Transformation, Math. Tables Aids Comput. 10 (1956), 91-96. https://doi.org/10.2307/2002183
  11. 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.
  12. A. Zygmund,Trigonometric Series, Second Edition, Cambridge University Press, Cambridge, 1959.