Lagrange Polynomial Approximation

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Let $f: D \to \R$ be $n+1$ times differentiable in an interval $I \subset \R$.

Let $x_0, \dotsc, x_n \in I$ be pairwise distinct points.

Let $P$ be the Lagrange Interpolation Formula of degree at most $n$ such that $P \left({x_i}\right) = f \left({x_i}\right)$ for all $i = 0, \dotsc, n$.

Let $R \left({x}\right) = f \left({x}\right) - P \left({x}\right)$.

Then, for every $x \in I$, there exists $\xi$ in the interval spanned by $x$ and $x_i$, $i = 0, \dotsc, n$, such that:

$R \left({x}\right) = \dfrac{\left({x - x_0}\right) \left({x - x_1}\right) \dotsm \left({x - x_n}\right) f^{\left({n + 1}\right)} \left({\xi}\right)} {\left({n + 1}\right)!}$


This proof is similar to the proof of Taylor's theorem with the remainder in the Lagrange form, and is also based on Rolle's Theorem.

Observe that:

$R \left({x_i}\right) = 0$ for $i = 0, \dotsc, n$

and that:

$R^{\left({n + 1}\right)} = f^{\left({n + 1}\right)}$

Without loss of generality, assume that $x$ is different from all $x_i$ for $i = 0, \dotsc, n$.

Let the function $g$ be defined by:

$g \left({t}\right) = R \left({t}\right) - \dfrac{\left({t - x_0}\right) \left({t - x_1}\right) \dotsm \left({t - x_n}\right) R \left({x}\right)} {\left({x - x_0}\right) \left({x - x_1}\right) \dotsm \left({x - x_n}\right)}$

Then $g \left({x_i}\right) = 0$ for $i = 0, \dotsc, n$, and $g \left({x}\right) = 0$.

Denote by $J$ the interval spanned by $x$ and $x_i$, $i = 0, \dotsc, n$.

Thus $g$ has at least $n + 2$ zeros in $J$.

The Extended Rolle's Theorem is applied in $J$ successively to $g$, $g'$, $g''$ and so on until $g^{\left({n + 1}\right)}$, which thus has at least one zero $\xi$ located between the two known zeros of $g^{\left({n}\right)}$ in $J$.


$0 = g^{\left({n + 1}\right)} \left({\xi}\right) = f^{\left({n + 1}\right)} \left({\xi}\right) - \dfrac{\left({n + 1}\right)! R \left({x}\right)} {\left({x - x_0}\right) \left({x - x_1}\right) \dotsm \left({x - x_n}\right)}$

and the formula for $R \left({x}\right)$ follows.



This theorem gives an estimate for the error of the Lagrange polynomial approximation and is similar to the Mean Value Theorem and Taylor's Theorem with the remainder in Lagrange form.