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Abstract.A criterion for the positivity of a cubic polynomial on a given interval is derived. By means of this result a necessary and sufficient condition is given under which cubic Ct-spline interpolants are nonnegative. Further, since such interpolants are not uniquely determined, for selecting one of them the geometric curvature is minimized. The arising optimization problem is solved numerically via dualization. AMS(MOS) classification:65D17, 41A15, 90C20.

--ZusammenfassungConvex Spline Interpolants with Minimal Curvature. The problem of finding convex spline interpolants with minimal mean curvature leads to a quadratic optimization problem of special structure. In the present note a corresponding dual problem without constraints is derived. Its objective function is piecewise quadratic and therefore admits an effective numerical treatment. . The problem to construct a cubic spline interpolant which is convex if the given data set is of this type may be not solvable. If a solution exists the convex spline interpolant is in general not uniquely determined. In this case it is appropriate to select that spline which has minimal curvature. These requirements lead to a quadratic programming problem of special structure. In the present note a corresponding dual problem without constraints is stated having the advantage that it can be solved effectively. The main result representing smooth duality statements is theorem 7. A MS Subject 2.In order to formulate the situation in more detail denote by A a grid on the interval

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