A Note on the Convergence of Interpolatory Cubic Splines

It is shown that if $x \in C^4 [a,b]$ is approximated by a natural cubic spline, then the error is $O(h^4 )$ 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 $O(h^4 )$ in $[a,b]$.