Hermite's Interpolating Polynomial

Let  be an th degree polynomial with zeros at , ..., . Then the fundamental Hermite interpolating polynomials of the first and second kinds are defined by

(1)

and

(2)

for , 2, ..., where the fundamental polynomials of Lagrange interpolation are defined by

(3)

They are denoted  and , respectively, by Szegö (1975, p. 330).

These polynomials have the properties

			(4)
			(5)
			(6)
			(7)

for , 2, ..., . Now let , ...,  and , ...,  be values. Then the expansion

(8)

gives the unique Hermite interpolating fundamental polynomial for which

			(9)
			(10)

If , these are called Hermite's interpolating polynomials.

The fundamental polynomials satisfy

(11)

and

(12)

Also, if  is an arbitrary distribution on the interval , then

			(13)
			(14)
			(15)
			(16)
			(17)
			(18)

where  are Christoffel numbers.

REFERENCES:

Bartels, R. H.; Beatty, J. C.; and Barsky, B. A. "Hermite and Cubic Spline Interpolation." Ch. 3 in An Introduction to Splines for Use in Computer Graphics and Geometric Modelling. San Francisco, CA: Morgan Kaufmann, pp. 9-17, 1998.

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.ش