Lagrange Interpolation Optimality

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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:

$\displaystyle \left.\frac{d^m E(e^{j\omega})}{d\omega^m}\right\vert _{\omega=0} = 0, \quad m=0,1,2,\ldots,N,$
where

$\displaystyle E(e^{j\omega})\isdef e^{-j\omega\Delta} - \sum_{n=0}^N h(n)e^{-j\omega n}$
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

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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