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^Nd"j^(^m^)u(x"j), where the x"j 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>=a"F"D where a"F"D=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

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