Lagrange Interpolation Optimality

In the uniformly sampled case, Lagrange interpolation can be viewed as ordinary FIR filtering:

• Lagrange interpolation filters maximally flat in the frequency domain about dc:
  $$\left.\frac{d^m E(e^{j\omega})}{d\omega^m}\right\vert _{\omega=0} = 0, \quad m=0,1,2,\ldots,N,$$
  where
  $$E(e^{j\omega})\isdef e^{-j\omega\Delta} - \sum_{n=0}^N h(n)e^{-j\omega n}$$
  and $\Delta$ is the desired delay in samples.
• This is the same optimality criterion as Butterworth filters in classical analog filter design

• Also can be viewed as ``Pade approximation'' to a constant frequency response in the frequency domain