Lagrange's Interpolating Fundamental Polynomial

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

 (1)

They have the property
 (2)

where is the Kronecker Delta. Now let , ..., be values. Then the expansion
 (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
 (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