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单词 Hermite's Interpolating Polynomial
释义

Hermite's Interpolating Polynomial

Let be an th degree Polynomial with zeros at , ..., . Then the fundamentalPolynomials are

(1)

and
(2)

for , 2, .... These polynomials have the properties
(3)
(4)
(5)
(6)

for , 2, ..., . Now let , ..., and , ..., be values. Then the expansion
(7)

gives the unique Hermite interpolating fundamental polynomial for which
(8)
(9)

If , these are called Step Polynomials. The fundamental polynomials satisfy
(10)

and
(11)

Also, if is an arbitrary distribution on the interval , then
(12)
(13)
(14)
(15)
(16)
(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.

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