Orthogonal Polynomials and Related Approximation Results

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


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.


Orthogonal Polynomial Polynomial Approximation Chebyshev Polynomial Legendre Polynomial Jacobi Polynomial 
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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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