Chebyshev method

The Fourier method is appropriate for periodic problems, but is not adapted to nonperiodic problems because of the existence of the Gibbs phenomenon at the boundaries. In the case of nonperiodic problems, it is advisable to have recourse to better-suited basis functions. Orthogonal polynomials, like Chebyshev polynomials, constitute a proper alternative to the Fourier basis. The Chebyshev series expansion may be seen as a cosine Fourier series, so that it possesses the valuable properties of the latter concerning, in particular, the convergence and the possible use of the FFT. On the other hand, the Chebyshev series expansion is exempt from the Gibbs phenomenon at the boundaries.