Table of Contents
Advances in Numerical Analysis
Volume 2016 (2016), Article ID 6758283, 10 pages
http://dx.doi.org/10.1155/2016/6758283
Research Article

Lebesgue Constant Using Sinc Points

1German University in Cairo, New Cairo City 11835, Egypt
2Faculty of Science, Ain Shams University, Cairo 11566, Egypt
3University of Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany

Received 20 April 2016; Revised 20 July 2016; Accepted 18 August 2016

Academic Editor: William J. Layton

Copyright © 2016 Maha Youssef et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. F. Stenger, M. Youssef, and J. Niebsch, “Improved approximation via use of transformations,” in Multiscale Signal Analysis and Modeling, X. Shen and A. I. Zayed, Eds., pp. 25–49, Springer, New York, NY, USA, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  2. L. Brutman, “On the Lebesgue function for polynomial interpolation,” SIAM Journal on Numerical Analysis, vol. 15, no. 4, pp. 694–704, 1978. View at Publisher · View at Google Scholar · View at MathSciNet
  3. L. N. Trefethen and J. A. C. Weideman, “Two results on polynomial interpolation in equally spaced points,” Journal of Approximation Theory, vol. 65, no. 3, pp. 247–260, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. A. Schönhage, “Fehlerfortpflanzung bei Interpolation,” Numerische Mathematik, vol. 3, pp. 62–71, 1961. View at Publisher · View at Google Scholar · View at MathSciNet
  5. C. Runge, “Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten,” Zeitschrift für Mathematik und Physik, vol. 46, pp. 224–243, 1901. View at Google Scholar
  6. G. Faber, “Uber die interpolatorische Darstellung stetiger Funktionen,” Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 23, pp. 192–210, 1914. View at Google Scholar
  7. S. Bernstein, “Sur l'ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné,” Mém. Cl. Sci. Acad. Roy. Belg, vol. 4, pp. 1–103, 1912. View at Google Scholar
  8. T. Rivlin, “The lebesgue constants for polynomial interpolation,” in Functional Analysis and Its Applications, H. Garnir, K. Unni, and J. Williamson, Eds., vol. 399 of Lecture Notes in Mathematics, pp. 422–437, Springer, Berlin, Germany, 1974. View at Google Scholar
  9. P. Erdős, “Problems and results on the theory of interpolation. II,” Acta Mathematica Academiae Scientiarum Hungarica, vol. 12, pp. 235–244, 1961. View at Google Scholar · View at MathSciNet
  10. C. De Boor and A. Pinkus, “Proof of the conjectures of Bernstein and Erdös concerning the optimal nodes for polynomial interpolation,” Journal of Approximation Theory, vol. 24, no. 4, pp. 289–303, 1978. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. F. Stenger, H. A. El-Sharkawy, and G. Baumann, “The Lebesgue constant for sinc approximations,” in New Perspectives on Approximation and Sampling Theory: Festschrift in Honor of Paul Butzer's 85th Birthday, A. I. Zayed and G. Schmeisser, Eds., Applied and Numerical Harmonic Analysis, chapter 13, pp. 319–335, Birkhäauser, Basel Switzerland, 2014. View at Publisher · View at Google Scholar
  12. J.-P. Berrut, “Rational functions for guaranteed and experimentally well-conditioned global interpolation,” Computers & Mathematics with Applications, vol. 15, no. 1, pp. 1–16, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. L. Bos, S. De Marchi, and K. Hormann, “On the Lebesgue constant of Berrut's rational interpolant at equidistant nodes,” Journal of Computational and Applied Mathematics, vol. 236, no. 4, pp. 504–510, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. R.-J. Zhang, “An improved upper bound on the Lebesgue constant of Berrut's rational interpolation operator,” Journal of Computational and Applied Mathematics, vol. 255, pp. 652–660, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. E. V. Strelkova and V. T. Shevaldin, “On Lebesgue constants of local parabolic splines,” Proceedings of the Steklov Institute of Mathematics, vol. 289, supplement 1, pp. 192–198, 2015. View at Publisher · View at Google Scholar
  16. S. Ghili and G. Iaccarino, “Reusing Chebyshev points for polynomial interpolation,” Numerical Algorithms, vol. 70, no. 2, pp. 249–267, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, Cambridge, UK, 1981. View at MathSciNet
  18. C. Schneider and W. Werner, “Some new aspects of rational interpolation,” Mathematics of Computation, vol. 47, no. 175, pp. 285–299, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J.-P. Berrut and H. D. Mittelmann, “Matrices for the direct determination of the barycentric weights of rational interpolation,” Journal of Computational and Applied Mathematics, vol. 78, no. 2, pp. 355–370, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. M. S. Floater and K. Hormann, “Barycentric rational interpolation with no poles and high rates of approximation,” Numerische Mathematik, vol. 107, no. 2, pp. 315–331, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. J. M. Carnicer, “Weighted interpolation for equidistant nodes,” Numerical Algorithms, vol. 55, no. 2-3, pp. 223–232, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. H. R. Schwarz, Numerische Mathematik, Teubner, Stuttgart, Germany, 4th edition, 1997, English Translation: Numerical Analysis: A Comprehensive Introduction, John Wiley & Sons, New York, NY, USA, 2nd edition, 1989.
  23. P. Henrici, Essentials of Numerical Analysis, John Wiley & Sons, New York, NY, USA, 1982. View at MathSciNet
  24. L. Bos, S. De Marchi, K. Hormann, and J. Sidon, “Bounding the Lebesgue constant for Berrut's rational interpolant at general nodes,” Journal of Approximation Theory, vol. 169, pp. 7–22, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. B. Fornberg, A Practical Guide to Pseudospectral Methods, vol. 1 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  26. G. Klein, “An extension of the Floater-Hormann family of barycentric rational interpolants,” Mathematics of Computation, vol. 82, no. 284, pp. 2273–2292, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. S. J. Smith, “Lebesgue constants in polynomial interpolation,” Annales Mathematicae et Informaticae, vol. 33, pp. 109–123, 2006. View at Google Scholar · View at MathSciNet
  28. F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  29. F. Stenger, Handbook of Sinc Methods, CRC Press, New York, NY, USA, 2011.