Hermite's Interpolating Polynomial

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

$$h^{(1)}_\nu(x)=\left[{1-{l''(x_\nu)\over l'(x_\nu)}}\right][l_\nu(x)]^2$$ (1)

and

$$h^{(2)}_\nu(x)=(x-x_\nu)[l_\nu(x)]^2$$ (2)

for $\nu=1$, 2, ...$n$. These polynomials have the properties

$$h^{(1)}_\nu(x_\mu) = \delta_{\nu\mu}$$ (3)

$${h^{(1)}_\nu}'(x_\mu) = 0$$ (4)

$$h^{(2)}_\nu(x_\mu) = 0$$ (5)

$${h^{(2)}_\nu}'(x_\mu) = \delta_{\nu\mu}.$$ (6)

for $\mu,\nu=1$, 2, ..., $n$. Now let $f_1$, ..., $f_n$ and $f_1'$, ..., $f_n'$ be values. Then the expansion

$$W_n(x)=\sum_{\nu=1}^n f_\nu h^{(1)}_\nu(x)+\sum_{\nu=1}^n f_\nu'h^{(2)}_\nu(x)$$ (7)

gives the unique Hermite interpolating fundamental polynomial for which

$$W_n(x_\nu) = f_\nu$$ (8)

$$W_n'(x_\nu) = f_\nu'.$$ (9)

If $f_\nu'=0$, these are called Step Polynomials. The fundamental polynomials satisfy

$$h_1(x)+\ldots+h_n(x)=1$$ (10)

and

$$\sum_{\nu=1}^n x_\nu h^{(1)}_\nu(x)+\sum_{\nu=1}^n h^{(2)}_\nu(x)=x.$$ (11)

Also, if $d\alpha(x)$ is an arbitrary distribution on the interval $[a,b]$, then

$$\int_a^b h^{(1)}_\nu(x)\,d\alpha(x) = \lambda_\nu$$ (12)

$$\int_a^b {h^{(1)}_\nu}'(x)\,d\alpha(x) = 0$$ (13)

$$\int_a^b x{h^{(1)}_\nu}'(x)\,d\alpha(x) = 0$$ (14)

$$\int_a^b h^{(2)}_\nu(x)\,d\alpha(x) = 0$$ (15)

$$\int_a^b {h^{(2)}_\nu}'(x)\,d\alpha(x) = \lambda_\nu$$ (16)

$$\int_a^b x{h^{(2)}_\nu}'(x)\,d\alpha(x) = \lambda_\nu x_\nu,$$ (17)

where $\lambda_\nu$ 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.