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Difference between revisions of "Christoffel-Darboux formula"

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''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" />''
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{{TEX|done}}
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''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
 
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>
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$$\sum_{k=0}^nP_k(x)P_k(t)=$$
  
<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>
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$$=\frac{\mu_n}{\mu_{n+1}}\frac{P_{n+1}(x)P_n(t)-P_n(x)P_{n+1}(t)}{x-t},$$
  
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
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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 [[#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.
 
extended the formula to arbitrary weight functions.

Latest revision as of 11:53, 19 November 2018

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
How to Cite This Entry:
Christoffel-Darboux formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Christoffel-Darboux_formula&oldid=43443
This article was adapted from an original article by P.K. Suetin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article