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Lagrange's Interpolating Fundamental Polynomial

Let $l(x)$ be an $n$th degree Polynomial with zeros at $x_1$, ..., $x_n$. Then the fundamental Polynomials are

l_\nu(x)={l(x)\over l'(x_\nu)(x-x_\nu)}.
\end{displaymath} (1)

They have the property
\end{displaymath} (2)

where $\delta_{\nu\mu}$ is the Kronecker Delta. Now let $f_1$, ..., $f_n$ be values. Then the expansion
L_n(x)=\sum_{\nu=1}^n f_\nu l_\nu(x)
\end{displaymath} (3)

gives the unique Lagrange Interpolating Polynomial assuming the values $f_\nu$ at $x_\nu$. Let $d\alpha(x)$ be an arbitrary distribution on the interval $[a,b]$, $\{p_n(x)\}$ the associated Orthogonal Polynomials, and $l_1(x)$, ..., $l_n(x)$ the fundamental Polynomials corresponding to the set of zeros of $p_n(x)$. Then
\int_a^b l_\nu(x)l_\mu(x)\,d\alpha(x)=\lambda_\mu\delta_{\nu\mu}
\end{displaymath} (4)

for $\nu, \mu=1$, 2, ..., $n$, where $\lambda_\nu$ are Christoffel Numbers.


Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 329 and 332, 1975.

© 1996-9 Eric W. Weisstein