Lagrange Interpolation Optimality

In the uniformly sampled case, Lagrange interpolation can be viewed as an FIR filter with coefficients that depend on the fractional part of the ``interpolation time'' (sample times on the integers)

• Lagrange interpolation filters are maximally flat in the frequency domain about dc:
  $$\left.\frac{d^m E(e^{j\omega})}{d\omega^m}\right\vert _{\omega=0} = 0, \quad m=0,1,2,\ldots,N,$$
  (1st $N+1$ terms of Taylor expansion about $\omega=0$ zeroed), where
  $$E(e^{j\omega})\mathrel{\stackrel{\mathrm{\Delta}}{=}}e^{-j\omega\Delta} - \sum_{n=0}^N h_\Delta(n)e^{-j\omega n}$$
  and $\Delta$ is the desired delay in samples
  (see text for proof)

• Same optimality criterion as Butterworth filters in classical analog filter design

• Can also be viewed as ``Padé approximation'' to a constant frequency response

• Also bounded by 1 in the frequency domain:
  $$\vert H(e^{j\omega T})\vert\le 1, \quad \forall\, \omega T \in [-\pi,\pi]$$