= Lloyd N. Trefethen
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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>.<Lloyd N. Trefethen homepage>[Appointments and book links]{ref}<Spectral Methods in MATLAB by Lloyd N. Trefethen>[Author's book page: forty programs and demonstrations of spectral accuracy]{ref}
\Image[https://people.maths.ox.ac.uk/trefethen/photo12small.jpg]
{title=Lloyd N. Trefethen}
{description=Portrait credited to Sarah Bird on <Lloyd N. Trefethen homepage>.}
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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.<Spectral Methods in MATLAB by Lloyd N. Trefethen>[Author's description of programs comparing spectral accuracy for varying smoothness]{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.
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