A Class of Cubic Splines Obtained Through Minimum Conditions

A class of cubic spline minimizing some special functional is investigated. This class is determined by the solution of a quadratic programming problem in which the minimizing function depends linearly on a parameter $\alpha < 2$. For $\alpha = 1/2$ natural splines are obtained. For $\alpha = - 1$ the spline minimizing the mean value of the third derivative is obtained. It is shown that this spline has the best convergence order.