Error Bounds for Interpolating Cubic Splines Under Various End Conditions

Conditions are given when some finite difference type operators acting on the second derivative of a C²-cubic spline interpolate of a function f over a locally uniform partition approximate $f''$, $f'''$ and f^iv at selected knots by orders up to O(h⁴ ). Points are identified where $f'$, $f''$ and $f'''$ are approximated by $s'$, $s''$ and $s'''$ to the order h⁴, h³ and h² respectively. End conditions are analyzed which give these results globally over uniform partitions for sufficiently smooth functions.