### 8.1 Polynomial interpolation

Although interpolation is not strictly a finite differencing topic, we briefly present it here because it is used below and in Section 9, when discussing spectral methods.

Given a set of distinct points (sometimes referred to as nodal points or nodes) and arbitrary associated function values , the interpolation problem amounts to finding (in this case) a polynomial of degree less than or equal to such that for .

It can be shown that there is one and only one such polynomial. Existence can be shown by explicit construction: suppose one had

where, for each , is a polynomial of degree less than or equal to such that
Then as given by Eq. (8.1) would interpolate at the nodal points . The Lagrange polynomials, defined as
indeed do satisfy Eq. (8.2). Uniqueness of the interpolant can be shown by using the property that polynomials of order can have at most roots, applied to the difference between any two interpolants.

Defining the interpolation error by

and assuming that is differentiable enough, it can be seen that satisfies
where is called the nodal polynomial of degree , and is in the smallest interval containing and . In other words, if we assume the ordering , then can actually be outside . For example, if , then . Sometimes, approximating by when is called extrapolation, and interpolation only if , even though an interpolating polynomial is used as approximation.