# Christoffel-Darboux formula

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

 $\sum_{k=0}^{n}\phi_{k}(x)\phi_{k}(y)={k_{n}\over k_{n+1}}\left({\phi_{n}(y)% \phi_{n+1}(x)-\phi_{n}(x)\phi_{n+1}(y)\over x-y}\right)$

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 $\{\phi_{i}\}_{i=0}^{n}$.

Title Christoffel-Darboux formula ChristoffelDarbouxFormula 2013-03-22 16:20:32 2013-03-22 16:20:32 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Theorem msc 42C05 msc 33D45