Christoffel-Darboux formula

Let ${\{{\varphi }_i\}}_{i=0}^n$ be orthonormal polynomials (the degree of ${\varphi }_k$ is $k$) and let $k_n$ be the coefficient of $x^n$ in ${\varphi }_n$. Then

$$\sum _{k=0}^n{\varphi }_k(x){\varphi }_k(y)=\frac{k_n}{k_{n+1}}\left(\frac{{\varphi }_n(y){\varphi }_{n+1}(x)-{\varphi }_n(x){\varphi }_{n+1}(y)}{x-y}\right)$$

The reason this formula is interesting is that the left-hand side is the integral kernel for the projection operator to the subspace spanned by the polynomials ${\{{\varphi }_i\}}_{i=0}^n$.

Title	Christoffel-Darboux formula
Canonical name	ChristoffelDarbouxFormula
Date of creation	2013-03-22 16:20:32
Last modified on	2013-03-22 16:20:32
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	9
Author	rspuzio (6075)
Entry type	Theorem
Classification	msc 42C05
Classification	msc 33D45