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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.
Thomas R. Lucas (Sat,) studied this question.