# 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 |