# proof of uniqueness of Lagrange Interpolation formula

Existence is clear from the construction, the uniqueness is proved by assuming there are two different polynomials^{} $p(x)$ and $q(x)$ that interpolate the points. Then $r(x)=p(x)-q(x)$ has $n$ zeros, ${x}_{1},\mathrm{\dots},{x}_{n}$ and there is a point ${x}_{e}$ such that $r({x}_{e})\ne 0$. $r(x)$ is non-constant with degree $\mathrm{deg}(r(x))\le n-1$ and has more than $n-1$ solutions, which is a contradiction^{}. Thus there can only be one polynomial.

Title | proof of uniqueness of Lagrange Interpolation formula |
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Canonical name | ProofOfUniquenessOfLagrangeInterpolationFormula |

Date of creation | 2013-03-22 14:09:25 |

Last modified on | 2013-03-22 14:09:25 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 10 |

Author | rspuzio (6075) |

Entry type | Proof |

Classification | msc 65D05 |

Classification | msc 41A05 |