Let
be an
th degree Polynomial with zeros at
, ...,
. Then the fundamental
Polynomials are
![\begin{displaymath}
l_\nu(x)={l(x)\over l'(x_\nu)(x-x_\nu)}.
\end{displaymath}](l1_87.gif) |
(1) |
They have the property
![\begin{displaymath}
l_\nu(x_\mu)=\delta_{\nu\mu},
\end{displaymath}](l1_88.gif) |
(2) |
where
is the Kronecker Delta.
Now let
, ...,
be values. Then the expansion
![\begin{displaymath}
L_n(x)=\sum_{\nu=1}^n f_\nu l_\nu(x)
\end{displaymath}](l1_92.gif) |
(3) |
gives the unique Lagrange Interpolating Polynomial assuming the values
at
. Let
be an arbitrary distribution on the interval
,
the associated Orthogonal
Polynomials, and
, ...,
the fundamental Polynomials corresponding to the
set of zeros of
. Then
![\begin{displaymath}
\int_a^b l_\nu(x)l_\mu(x)\,d\alpha(x)=\lambda_\mu\delta_{\nu\mu}
\end{displaymath}](l1_101.gif) |
(4) |
for
, 2, ...,
, where
are Christoffel Numbers.
References
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI:
Amer. Math. Soc., pp. 329 and 332, 1975.
© 1996-9 Eric W. Weisstein
1999-05-26