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On monotone and convex spline interpolation. (English) Zbl 0617.41015
The author investigates the problem of the existence of monotone and/or convex splines which have degree n and order of continuity k and which interpolate a given set of data. Where they exist, the author gives a method for obtaining the desired splines by using Bernstein polynomials of appropriate piecewise linear functions. A number of algorithms involving the given data are developed which indicate the existence or non-existence of a solution to the problem with appropriate monotone or convex characteristics, or both. In all cases considered, a necessary condition for a solution to exist is that $k\le n-k$. These results generalize results due to D. F. McAllister, E. Passow and J. A. Roulier [Math. Comput. 31, 717-725 (1977; Zbl 0371.65001)] and E. Passow and J. A. Roulier [SIAM J. Numer. Anal. 14, 904-909 (1977; Zbl 0378.41002)].
Reviewer: P.Lappan

##### MSC:
 41A15 Spline approximation
##### Keywords:
monotone spline; convex splines; Bernstein polynomials