Christoffel-Darboux formula

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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.

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
How to Cite This Entry:
Christoffel-Darboux formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Christoffel-Darboux_formula&oldid=17001
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article