Orthogonal systems in L 2 .R/, once implemented in spectral methods, enjoy a number of important advantages if their differentiation matrix is skew-symmetric and highly structured. Such systems, where the differentiation matrix is skewsymmetric, tridiagonal, and irreducible, have been recently fully characterised. In this paper we go a step further, imposing the extra requirement of fast computation: specifically, that the first N coefficients of the expansion can be computed to high accuracy in O.N log 2 N / operations. We consider two settings, one approximating a function f directly in . I; I/ and the other approximating f .x/ g f . x/=2 and f .x/ f . x/=2 separately in 0; I/. In each setting we prove that there is a single family, parametrised by ; > 1, of orthogonal systems with a skew-symmetric, tridiagonal, irreducible differentiation matrix and whose coefficients can be computed as Jacobi polynomial coefficients of a modified function. The four special cases where ; h ¦1=2 are of particular interest, since coefficients can be computed using fast sine and cosine transforms. Banded, Toeplitz-plus-Hankel multiplication operators are also possible for representing variable coefficients in a spectral method. In Fourier space these orthogonal systems are related to an apparently new generalisation of the Carlitz polynomials.