Interpolation

  

Figure: Polynomial Interpolation

Definition 1684

A polynomial interpolates a given function if it takes the same values as the function at a number of points, called the interpolation points.

We can draw a straight line through two given interpolation points, a parabola through three and an nth degree polynomial through n+1 points. We prove that this is always possible provided that the points are distinct.

Theorem 1694

Given n+1 distinct points $x_0,x_1,\ldots ,x_n$ and corresponding values $y_0,y_1,\ldots ,y_n$ there exists a unique polynomial p of degree n in x for which $p(x_i)=y_i,~i=0,1,\ldots ,n$.

Note that we have only specified values here; these might be the values of a given function at the interpolation points, but they need not be for the theorem to be valid.

This result is not specific to polynomials; for example, we can use the interpolating function $\sum _{r=0}^n(a_r\cos rx+b_r\sin rx)$. Here we have 2n+1 parameters ($\sin 0x=0$), so that we should expect to be able to interpolate at 2n+1 points, $x_0,x_1,\ldots ,x_{2n}$, say. The interpolation conditions now give the determinant

$$\left\vert \begin{array} {cccccc} 1 & \cos x_0 & \sin x_0 & ... ... \cdots & \cos nx_{2n} & \sin nx_{2n} \end{array} \right\vert,$$

which can be shown to be nonzero for distinct points, thus giving a unique solution.

Both this and the polynomial case are special cases of a general result for interpolating functions which form a Chebyshev set, but this is a subject for study in Approximation Theory.