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

Recently, Boyd published a paper entitled ‘‘A proof that the discrete singular convolution (DSC)/ Lagrange-distributed approximating function (LDAF) method is inferior to high order finite differences’’ (J. Comput. Phys. 214 (2006) 538–549 [3]), which will be referred to as ‘‘Proof’’. This Letter analyzes the validity of central claims in ‘‘Proof’’. In its title and abstract, ‘‘Proof’’ refers to LDAF as ‘‘Lagrange-distributed approximating function’’, while in its introduction, ‘‘Proof’’ refers to LDAF as ‘‘linear distributed approximating functional’’ and attributes it to Hoffman et al. [4]. In fact, ‘‘linear distributed approximating functional (LDAF)’’ does not exist. The paper by Hoffman et al. [4] was concerned the ‘‘distributed approximating functional’’. ‘‘Proof’’ did not give any detailed analysis, theoretical expression, and correct literature reference about LDAF. Moreover, the DSC algorithm can be realized by many different kernels that may behave very differently from each other [8,9]. What is discussed in ‘‘Proof’’ is a special case, the regularized Shannon kernel (RSK) dðx xjÞ 1⁄4 sinhðx xjÞ p hðx xjÞ expð a1⁄2x xj =hÞ, where h is the grid spacing. We therefore limit our discussion to the DSCRSK method. First, ‘‘Proof’’ states in its abstract that ‘‘we show that the DSC is worse than the standard finite differences in differentiating exp for all k when a P aFD where aFD 1⁄4 1= ffiffiffiffiffiffiffiffiffiffiffiffi N þ 1 p with N as the stencil width is the value of the DSC parameter that makes its weights most closely resemble those of finite differences’’. Because a is a free parameter, no practitioner would impose the condition a P aFD, but instead, an optimal choice for this parameter is sought and this can be a value a < aFD. Second, ‘‘Proof’’ states in its abstract that ‘‘For a < aFD, the DSC errors are less than finite differences for k near the aliasing limit, but much, much worse for smaller k’’. Although here ‘‘Proof’’ does not specify what is meant by ‘‘small k’’ and what is the remainder, it gives two intervals jKj < p 2 and jKj > p 2 in Section 6, where K = kh. In Fig. 1a, a counterexample is given to show that the DSC method outperforms the finite difference