# 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, $x_{1},\ldots,x_{n}$ and there is a point $x_{e}$ such that $r(x_{e})\neq 0$. $r(x)$ is non-constant with degree $\deg(r(x))\leq n-1$ 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 ProofOfUniquenessOfLagrangeInterpolationFormula 2013-03-22 14:09:25 2013-03-22 14:09:25 rspuzio (6075) rspuzio (6075) 10 rspuzio (6075) Proof msc 65D05 msc 41A05