Chapter 3.5, Problem 33ES

Let f ∈ C²[a, b] and let the nodes a = x₀ < x₁ < ⋯ < x_n = b be given. Derive an error estimate similar to that in Theorem 3.13 for the piecewise linear interpolating function F. Use this estimate to derive error bounds for Exercise 15.

Theorem 3.13 Let f ∈ C⁴[a, b] with max_a≤x≤b |f⁽⁴⁾(x)| = M. If S is the unique clamped cubic spline interpolant to f with respect to the nodes a = x₀ < x₁ < ⋯ < x_n = b, then, for all x in [a, b].

$|f(x)-S(x)|\le \frac{5M}{384}\underset{0\le j\le n-1}{\max }{\left(x_{j+1}-x_j\right)}^4.$