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Lloyd N. Trefethen is a mathematician working in numerical analysis. His author-maintained homepage records work at Oxford and a subsequent applied-mathematics appointment at Harvard. His book Spectral Methods in MATLAB by Lloyd N. Trefethen teaches spectral methods through short executable programs, connecting approximation accuracy with numerical solutions of differential equations.^[ref][ref]

An instructive theme is that an approximation's smoothness affects how efficiently its coefficients decay. A Fourier series representation of a smooth periodic function can be differentiated through its coefficients, but unresolved oscillations can still produce aliasing. High accuracy for one smooth test function does not establish that every sampled input is equally well represented.^[ref]

For a concrete illustration, sampling $\sin (3x)$ on sufficiently many equally spaced points over a full period permits its trigonometric interpolant to reproduce the function exactly, and coefficient differentiation gives $3\cos (3x)$. With only four equally spaced points, the samples instead match $-\sin x$; differentiating that interpolant gives $-\cos x$. The calculation illustrates the attraction of spectral methods and the need to state their resolution assumptions. It is an original example of the numerical questions made tangible by his computational exposition.

A pseudospectral method represents an approximate solution through a global basis or its interpolating values and imposes equations at selected collocation points. In a periodic problem the basis is often a Fourier series; on an interval, polynomial collocation is common. The adjective spectral refers to the representation, not to computing the eigenvalues of an arbitrary matrix.^[ref]

For periodic data at $x_j=2\pi j/N$, the discrete Fourier transform gives coefficients ${\hat{u}}_k$ of a trigonometric interpolant. Its derivative is obtained by multiplying coefficient $k$ by $ik$ and transforming back. For nonlinear equations, products can be evaluated on the physical grid, while derivatives are computed in Fourier coordinates. A Fast Fourier transform makes these representation changes efficient.

The grid can confuse unresolved frequencies with resolved ones. With $N=8$, take $u(x)=\cos (3x)$. Its exact square isAt the eight grid points, $\cos (6x_j)=\cos (2x_j)$, so the sampled product appears to contain wave number $2$ instead of $6$. This is aliasing created by the nonlinear product, even though the input wave number $3$ was representable. Additional grid resolution, padding or appropriate filtering is needed when such products are used to evolve retained modes.

For sufficiently smooth solutions, spectral representations can converge rapidly as resolution increases. Discontinuities, boundary incompatibilities and unresolved fine structure change that behavior. A pseudospectral method therefore needs checks on both approximation error and the nonlinear operations performed on its grid.