End conditions for cubic spline interpolation derived from integration (Q1119847)

End conditions are considered, which determine an interpolatory cubic spline uniquely and in a way, that \(O(h^ 4)\) approximation to \(C^ 4\) functions is achieved. The author surveys numerous known conditions and gives new ones which involve function values in knots (data) only and are obtained from formulas of numerical integration: the integral of the spline over an interval is expressed in terms of values and derivatives at the endpoints. Equating this to the value obtained by an integration formula for the measurements, yields the required conditions, which contain some known conditions as special cases.