View source for Christoffel-Darboux formula

''for polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022190/c0221901.png" /> that are orthonormal with an integral weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022190/c0221902.png" /> on some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022190/c0221903.png" />''

A formula of the type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022190/c0221904.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022190/c0221905.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022190/c0221906.png" /> is the leading coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022190/c0221907.png" />. The Christoffel–Darboux formula is used in investigating conditions of convergence for Fourier series in orthogonal polynomials at a single point. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022190/c0221908.png" /> is a step function, the Christoffel–Darboux formula was first published by P.L. Chebyshev in 1855 (see [[#References|[1]]]). E.B. Christoffel [[#References|[2]]] then established it for the [[Legendre polynomials|Legendre polynomials]], and G. Darboux

extended the formula to arbitrary weight functions.

See also the references to [[Orthogonal polynomials|Orthogonal polynomials]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.L. Chebyshev,  , ''Collected works'' , '''2''' , Moscow-Leningrad  (1947)  pp. 103–106  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.B. Christoffel,   "Ueber die Gausssche Quadratur und eine Verallgemeinerung derselben"  ''J. Reine Angew. Math.'' , '''55'''  (1858)  pp. 61–82</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  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</TD></TR></table>