Christoffel-Darboux formula

for polynomials that are orthonormal with an integral weight on some interval

A formula of the type

where is the leading coefficient of . The Christoffel–Darboux formula is used in investigating conditions of convergence for Fourier series in orthogonal polynomials at a single point. In case is a step function, the Christoffel–Darboux formula was first published by P.L. Chebyshev in 1855 (see [1]). E.B. Christoffel [2] then established it for the Legendre polynomials, and G. Darboux

extended the formula to arbitrary weight functions.

See also the references to Orthogonal polynomials.

References

[1]	P.L. Chebyshev, , Collected works , 2 , Moscow-Leningrad (1947) pp. 103–106 (In Russian)
[2]	E.B. Christoffel, "Ueber die Gausssche Quadratur und eine Verallgemeinerung derselben" J. Reine Angew. Math. , 55 (1858) pp. 61–82
[3a]	G. Darboux, "Mémoire sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série" J. Math. Pures Appl. (3) , 4 (1878) pp. 5–56
[3b]	G. Darboux, "Sur l'approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série" J. Math. Pures Appl. (3) , 4 (1878) pp. 377–416