Lloyd N. Trefethen

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.