Christoffel-Darboux formula

for polynomials $P_n(x)$ that are orthonormal with an integral weight $d\sigma(x)$ on some interval $(a,b)$

A formula of the type

$$\sum_{k=0}^nP_k(x)P_k(t)=$$

$$=\frac{\mu_n}{\mu_{n+1}}\frac{P_{n+1}(x)P_n(t)-P_n(x)P_{n+1}(t)}{x-t},$$

where $\mu_n$ is the leading coefficient of $P_n(x)$. The Christoffel–Darboux formula is used in investigating conditions of convergence for Fourier series in orthogonal polynomials at a single point. In case $\sigma(x)$ 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