In this paper, we construct discrete fractional Fourier transforms (DFrFT) using recently introduced closed-form Hermite–Gaussian-like (HGL) eigenvectors. With respect to such eigenvectors, we discuss the convergence of their components to samples of the corresponding continuous Hermite–Gaussian functions and propose solutions to deal with some restrictions related to their construction. This allows us to give new procedures for obtaining orthonormal bases of HGL eigenvectors, which are used to fractionalize the discrete Fourier transform. We illustrate the application of the resulting DFrFT in the scenario of filtering in the fractional domain and compare the results with existing DFrFT approaches.