Let $x_0, x_1, … , x_n$ be distinct real or complex numbers, and let $y_0 , y_1, …, y_n$ be associated function values. We now study the problem of finding a polynomial p (x) that interpolates the given data:

By writing: $p(x) = a_0 + a_1x + … + a_mx^m$

Consider that m = n, then

It can be written that $Xa = y$, with

The matrix X is called a Vandermonde matrix.

Theorem

Given n + 1 distinct points $x_0, … , x_n$ and n + 1 ordinates $y_0, …, y_n$, there is a polynomial p(x) of $degree \le n$ that interpolates $y_i$ at $x_i$, i = 0, 1, … , n. This polynomial p(x) is unique among the set of all polynomials of degree at most n.

Below are three kinds of proofs.

Proof

Reference