A Note on the Convergence of Interpolatory Cubic Splines

It is shown that if x ∈ C⁴ [a,b] is approximated by a natural cubic spline, then the error is O(h⁴ ) in a closed interval which is asymptotic to $[a,b]$ as h, the maximum interval length, decreases to zero. A by-product of the technique used is that if exact end conditions are imposed, then the error is O(h⁴ ) in $[a,b]$.