Certain Properties of Polynomials

Abstract

Let x ₁,…, x _n+1 be distinct points in the complex plane ℂ. Then there exists precisely one polynomial P(x) of degree not greater than n which takes a prescribed value a _i at x _i . Indeed, the uniqueness of P follows from the fact that the difference of two such polynomials vanishes at points x ₁,…, x _n+1 and at the same time has degree not greater than n. The following polynomial clearly possesses all the necessary properties:

$$\begin{array}{ll} P(x) & = \sum\limits_{k=1}^{n+1} a_k \frac{(x - x_1)\cdot\ldots\cdot(x - x_{k-1})\cdot(x - x_{k+1})\cdot\ldots\cdot(x - x_{n+1})}{(x_k - x_1)\cdot\ldots\cdot(x_k - x_{k-1})\cdot(x_k - x_{k+1})\cdot\ldots\cdot(x_k - x_{n+1})} =\\ & = \sum\limits_{k=1}^{n+1} a_k\frac{\omega(x)}{(x-x_k)\omega^\prime (x_k)},\end{array}$$

where

$$\omega(x) = (x - x_1)\cdot\ldots\cdot(x - x_{n+1}).$$

The polynomial P(x) is called Lagrange’s interpolation polynomial and the points x ₁,… , x _n+1 are called the interpolation nodes.