A proof that the discrete singular convolution (DSC)/Lagrange-distributed approximating function (LDAF) method is inferior to high order finite differences

Finite differences approximate the mth derivative of a function u(x) by a series ∑ j = - N N d j ( m ) u ( x j ) , where the xj are the grid points. The closely-related discrete singular convolution (DSC) and Lagrange-distributed approximating function (LDAF) methods, treated here as a single algorithm, approximate derivatives in the same way as finite differences but with different numerical weights that depend upon a free parameter a. By means of Fourier analysis and error theorems, we show that the DSC is worse than the standard finite differences in differentiating exp(ikx) for all k when a ⩾ aFD where a FD ≡ 1 / N + 1 with N as the stencil width is the value of the DSC parameter that makes its weights most closely resemble those of finite differences. For a < aFD, the DSC errors are less than finite differences for k near the aliasing limit, but much, much worse for smaller k. Except for the very unusual case of low-pass filtered functions, that is, functions with negligible amplitude in small wavenumbers k, the DSC/LDAF is less accurate than finite differences for all stencil widths N. So-called “spectrally-weighted” or “frequency-optimized” differences are superior for this special case. Consequently, DSC/LDAF methods are never the best way to approximate derivatives on a stencil of a given width.