Let
be an
th degree Polynomial with zeros at
, ...,
. Then the fundamental
Polynomials are
![\begin{displaymath}
h^{(1)}_\nu(x)=\left[{1-{l''(x_\nu)\over l'(x_\nu)}}\right][l_\nu(x)]^2
\end{displaymath}](h_1377.gif) |
(1) |
and
![\begin{displaymath}
h^{(2)}_\nu(x)=(x-x_\nu)[l_\nu(x)]^2
\end{displaymath}](h_1378.gif) |
(2) |
for
, 2, ...
. These polynomials have the properties
for
, 2, ...,
. Now let
, ...,
and
, ...,
be values. Then the expansion
![\begin{displaymath}
W_n(x)=\sum_{\nu=1}^n f_\nu h^{(1)}_\nu(x)+\sum_{\nu=1}^n f_\nu'h^{(2)}_\nu(x)
\end{displaymath}](h_1391.gif) |
(7) |
gives the unique Hermite interpolating fundamental polynomial for which
If
, these are called Step Polynomials. The fundamental polynomials satisfy
![\begin{displaymath}
h_1(x)+\ldots+h_n(x)=1
\end{displaymath}](h_1397.gif) |
(10) |
and
![\begin{displaymath}
\sum_{\nu=1}^n x_\nu h^{(1)}_\nu(x)+\sum_{\nu=1}^n h^{(2)}_\nu(x)=x.
\end{displaymath}](h_1398.gif) |
(11) |
Also, if
is an arbitrary distribution on the interval
, then
![$\displaystyle \int_a^b h^{(1)}_\nu(x)\,d\alpha(x)$](h_1401.gif) |
![$\textstyle =$](h_42.gif) |
![$\displaystyle \lambda_\nu$](h_1402.gif) |
(12) |
![$\displaystyle \int_a^b {h^{(1)}_\nu}'(x)\,d\alpha(x)$](h_1403.gif) |
![$\textstyle =$](h_42.gif) |
![$\displaystyle 0$](h_863.gif) |
(13) |
![$\displaystyle \int_a^b x{h^{(1)}_\nu}'(x)\,d\alpha(x)$](h_1404.gif) |
![$\textstyle =$](h_42.gif) |
![$\displaystyle 0$](h_863.gif) |
(14) |
![$\displaystyle \int_a^b h^{(2)}_\nu(x)\,d\alpha(x)$](h_1405.gif) |
![$\textstyle =$](h_42.gif) |
![$\displaystyle 0$](h_863.gif) |
(15) |
![$\displaystyle \int_a^b {h^{(2)}_\nu}'(x)\,d\alpha(x)$](h_1406.gif) |
![$\textstyle =$](h_42.gif) |
![$\displaystyle \lambda_\nu$](h_1402.gif) |
(16) |
![$\displaystyle \int_a^b x{h^{(2)}_\nu}'(x)\,d\alpha(x)$](h_1407.gif) |
![$\textstyle =$](h_42.gif) |
![$\displaystyle \lambda_\nu x_\nu,$](h_1408.gif) |
(17) |
where
are Christoffel Numbers.
References
Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp. 314-319, 1956.
Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 330-332, 1975.
© 1996-9 Eric W. Weisstein
1999-05-25