Motivated by the computation of quantum problems in semiclassical regime, we explore fast approximation of functions on the real line, in particular of wave packets – by "fast" we mean both rapid speed of convergence and the derivation of the first n expansion coefficients in O(n log n) operations. Specifically, we need to construct an orthogonal system in L(-∞,∞) with these welcome features and we explore four candidates: Hermite polynomials, Hermite functions, stretched Fourier functions and stretched Chebyshev polynomials. We analyse their speed of convergence, describe some unexpected phenomena and, using a panoply of techniques, determine the surprising winner.