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Lagrange Interpolation


Problem Formulation

Given a set of $ N+1$ known samples $ f(x_k)$ , $ k=0,1,2,\ldots,N$ , find the unique order $ N$ polynomial $ y(x)$ which interpolates the samples

Solution (Waring, Lagrange):

$\displaystyle y(x) = \sum_{k=0}^N l_k(x)f(x_k)
$

where

$\displaystyle l_k(x) \mathrel{\stackrel{\mathrm{\Delta}}{=}}{ (x - x_0) \cdots (x - x_{k-1}) (x - x_{k+1}) \cdots (x - x_N)
\over (x_k - x_0) \cdots (x_k - x_{k-1}) (x_k - x_{k+1}) \cdots (x_k - x_N) }
$



Subsections
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``Bandlimited Interpolation, Fractional Delay Filtering, and Optimal FIR Filter Design'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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