Table of Contents
What is Lagrange Interpolation Theorem?
The Lagrange Interpolation Theorem states that if a function is continuous on a closed interval and has a derivative on the interior of the interval, then there exists a polynomial that interpolates the function on the interval.
Interpolation
Interpolation is a mathematical method of estimating the value of a function at a point not given by an actual observation. The estimated value is obtained by assuming a smooth curve passes through the given points. The smooth curve is called an interpolating function.
Lagrange Interpolation Theorem
The Lagrange interpolation theorem states that if is a function defined on a set of points in , and is a function defined on a subset of , then there exists a function satisfying for all in .
Proof of Lagrange Theorem
Proof:
We will use the following lemma:
Lemma: Let $f$ be a function on a set $S$ and let $x$ be an element of $S$. If $f$ is continuous at $x$, then there exists a number $c$ such that $f(c) = x$.
Proof:
Let $f$ be a function on a set $S$ and let $x$ be an element of $S$. If $f$ is continuous at $x$, then there exists a number $c$ such that $f(c) = x$.
