  Many standard conversion matrices between coefficients in classical
orthogonal polynomial expansions can be decomposed using diagonally-scaled
Hadamard products involving Toeplitz and Hankel matrices. This allows us to
derive $\smash{\mathcal{O}(N(\log N)^2)}$ algorithms, based on the fast Fourier
transform, for converting coefficients of a degree $N$ polynomial in one
polynomial basis to coefficients in another. Numerical results show that this
approach is competitive with state-of-the-art techniques, requires no
precomputational cost, can be implemented in a handful of lines of code, and is
easily adapted to extended precision arithmetic.
