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.
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© 2011 Springer Berlin Heidelberg
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Shen, J., Tang, T., Wang, LL. (2011). Orthogonal Polynomials and Related Approximation Results. In: Spectral Methods. Springer Series in Computational Mathematics, vol 41. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71041-7_3
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DOI: https://doi.org/10.1007/978-3-540-71041-7_3
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Publisher Name: Springer, Berlin, Heidelberg
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Online ISBN: 978-3-540-71041-7
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