It is well-known that the discrete Fourier transform (DFT) of a finite length discrete-time signal samples the discrete-time Fourier transform of the same signal at equidistant points on the unit circle. Hence, as the signal length goes to infinity, the DFT approaches the DTFT. Associated with the DFT are circular convolution and a periodic signal extension. In this paper we identify a large class of alternatives to the DFT using the theory of polynomial algebras. Each of these Fourier transforms approaches the DTFT just as the DFT does, but has its own signal extension and notion of convolution, which therefore are not periodic. Furthermore, these Fourier transforms have Vandermonde structure, which enables their computation via fast $O(n \log^2(n))$ algorithms.