Comparison

In both the DVR and the discrete representation a discrete Hilbert space is employed. In the Fourier representation a uniform grid spacing is used to obtain a finite Fourier series. The quadrature weight is the grid spacing (trapezoidal rule). This quadrature is of comparable accuracy to the Gaussian quadrature employed in the DVR representation.

The Fourier method scales as $N\log N$ as opposed to an $N^{2}$ scaling of the DVR method. The DVR, however, provides the opportunity of choosing the basis functions $\phi _{j}$ according to the physical system so their number can be kept rather small (e.g. LCAO).