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Orthogonal Polynomials and Related Approximation Results

  • Jie ShenEmail author
  • Tao Tang
  • Li-Lian Wang
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 41)

Abstract

The Fourier spectral method is only appropriate for problems with periodic boundary conditions. If a Fourier method is applied to a non-periodic problem, it inevitably induces the so-called Gibbs phenomenon, and reduces the global convergence rate to O(N-1) (cf. Gottlieb and Orszag (1977)). Consequently, one should not apply a Fourier method to problems with non-periodic boundary conditions. Instead, one should use orthogonal polynomials which are eigenfunctions of some singular Sturm-Liouville problems. The commonly used orthogonal polynomials include the Legendre, Chebyshev, Hermite and Laguerre polynomials.

Keywords

Orthogonal Polynomial Polynomial Approximation Chebyshev Polynomial Legendre Polynomial Jacobi Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsHong Kong Baptist UniversityKowloonHong Kong SAR
  3. 3.Division of Mathematical Sciences School of Physical & Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

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