Algorithm xxx: The Matlab Postprocessing Toolkit

Spectral methods approximate functions by projection onto a space PN of orthogonal polynomials of degree ≤ N . When the underlying function is periodic trigonometric (Fourier) polynomials are employed while a popular choice for non-periodic functions are the Chebyshev polynomials. Legendre polynomials are another option in the non-periodic case but are not as popular in applications due to the lack of a fast transform method. However spectral methods based on other, nonclassical orthogonal polynomials are possible as well. For example see references [Chen et al. 2005] and [Weideman 1999]. Spectral methods yield extremely accurate approximations of smooth functions. Due to their excellent approximation properties, spectral methods have become popular in applications such as numerical partial differential equations. However, when functions are only piecewise smooth the well-known Gibbs phenomenon appears as an accuracy reduction to first order away from discontinuities andO (1) oscillations in the neighborhoods of jumps. The Gibbs phenomenon is illustrated in figure 1 for both the Fourier and Chebyshev approximation of the function