Math 6610 • Analysis of Numerical Methods I

{Interpolation} Suppose \(f\) is arbitrarily often differentiable and
let \(p\) be the unique polynomial of degree \(d\) satisfying
\[
  p(x_i)=f(x_i),\qquad i=0,\ldots,n
\]
where the abscissas \(x_i\) are distinct. Derive a suitable expression for the
error \(f(x)-p(x)\) and discuss its properties.