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Lagrange Interpolation

Lagrange interpolation is a well known, classical technique for interpolation [194]. It is also called Waring-Lagrange interpolation, since Waring actually published it 16 years before Lagrange [312, p. 323]. More generically, the term polynomial interpolation normally refers to Lagrange interpolation. In the first-order case, it reduces to linear interpolation.

Given a set of $ N+1$ known samples $ f(x_k)$ , $ k=0,1,2,\ldots,N$ , the problem is to find the unique order $ N$ polynomial $ y(x)$ which interpolates the samples.5.2The solution can be expressed as a linear combination of elementary $ N$ th order polynomials:

$\displaystyle y(x) = \sum_{k=0}^N l_k(x)f(x_k) \protect$ (5.6)

where

$\displaystyle l_k(x) \isdef \frac{(x - x_0) \cdots (x - x_{k-1}) (x - x_{k+1}) \cdots (x - x_N)
}{(x_k - x_0) \cdots (x_k - x_{k-1}) (x_k - x_{k+1}) \cdots (x_k - x_N)}.
$

From the numerator of the above definition, we see that $ l_k(x)$ is an order $ N$ polynomial having zeros at all of the samples except the $ k$ th. The denominator is simply the constant which normalizes to give $ l_k(x_k)=1$ . Thus, we have

$\displaystyle l_k(x_j) = \delta_{kj} \isdef \left\{\begin{array}{ll}
1, & j=k, \\ [5pt]
0, & j\neq k. \\
\end{array} \right.
$

The polynomial $ l_k$ can be interpreted as the $ k$ th basis polynomial for constructing a polynomial interpolation of order $ N$ over the $ N+1$ sample points $ x_k$ . It is an order $ N$ polynomial having zeros at all of the samples except the $ k$ th, where it is 1. An example of a set of eight basis functions $ l_k$ for randomly selected interpolation points $ x_k$ is shown in Fig.4.10.

Figure 4.10: Example Lagrange basis functions in the eighth-order case for randomly selected interpolation points (marked by dotted lines). The unit-amplitude points are marked by dashed lines.
\includegraphics[width=\twidth]{eps/lagrangebases}



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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