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In this paper we propose a method to construct shape-preserving C cubic polynomial splines interpolating convex and/or monotonie data. For such given data, the existence or nonexistence of such interpolating splines can be expressed in terms of existence or nonexistence of solutions for a system of linear inequalities in two unknowns. 0. Introduction In many interpolation problems it is important that the solution preserves some shape properties such as convexity or monotonicity. Classical methods (the polynomial spline functions being the most widely used) usually ignore these kinds of conditions and thus yield solutions exhibiting undesirable inflections or oscillations. This is the reason why many investigations during the last years have been directed towards interpolation by means of shape-preserving polynomial spline functions. In [6] McAllister and Roulier, and in [13] Schumaker, have studied quadratic splines which preserve monotonicity and convexity. In [5] Fritsch and Carlson have studied cubic splines that preserve monotonicity. In [1, 2] Costantini and Morandi have studied cubic splines which preserve both convexity and monotonicity. All of these splines are C1. Other authors (Neuman [10, 11] and Mettke [9]) have imposed additional conditions on the monotone, convex data, which yield a solution that belongs to a subspace of polynomial splines. Moreover, McAllister and Roulier [7], and Passow and Roulier [12] have shown that it may be impossible to construct monotonie and convex splines of given degree and deficiency. In [8] Medina gives a survey on shape-preserving interpolation by means of polynomial or of other classes of splines. In this paper we propose a method to construct C cubic polynomial interpolation splines. Functions of this kind, of course, do not always exist for arbitrary convex monotone data sets. For convex increasing data—this, as we Received July 12, 1989; revised January 18, 1990, May 17, 1990, July 25, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 65D07, 65D10, 41A05.

Shape-preserving $C\sp 2$ cubic polynomial interpolating splines