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Christoffel-Darboux Identity
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 sum_(k=0)^m(phi_k(x)phi_k(y))/(gamma_k)=(phi_(m+1)(x)phi_m(y)-phi_m(x)phi_(m+1)(y))/(a_mgamma_m(x-y),)
(1)

where phi_k(x) are orthogonal polynomials with weighting function W(x) such that

 gamma_m=int[phi_m(x)]^2W(x)dx,
(2)

and

 a_k=(A_(k+1))/(A_k)
(3)

with A_k is the coefficient of x^k in phi_k(x).

SEE ALSO: Christoffel Formula, Orthogonal Polynomials

REFERENCES:

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, p. 322, 1956.




CITE THIS AS:

Weisstein, Eric W. "Christoffel-Darboux Identity." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Christoffel-DarbouxIdentity.html

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