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Fast polynomial transforms based on Toeplitz and Hankel matrices

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.