Interpolation concerns the question of finding a function that has prescribed values at a given number of points. In the polynomial context we are of course looking for polynomial functions.

Given n points x1, ..., x n R, and n prescribed values a1, ..., an R, does a polynomial function f : R -> R exist that interpolates the values (ai) on (xi)?

Lagrange

Lagrange interpolation theorem

Let n be an integer and R a field. Suppose n distinct points x1, ..., xn R and n required values a1, ..., an R are given. Then there is a unique polynomial function f : R -> R of degree at most n - 1 with f(xi) = ai for all i.