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Convex interpolating splines of arbitrary degree. II. (English) Zbl 0559.41005

In this paper the author continues the investigations of splines with preassigned smoothness, started in his previous work published in the volume: ”Numerical Methods of Approximation Theory”, Vol. 5, 211-222 (1980; Zbl 0436.41001). One denotes by \(Sp(k+1,j,\Delta_ n)\) the set of polynomial splines of degree \(k+1\) and deficiency equal to \(k+1-j\), where \(k\in N_ 0,0\leq j\leq k\) and \(\Delta_ n:a=x_ 0<x_ 1<...<x_ n=b\). For fixed k and j (0\(\leq j\leq k\), \(k\geq 1)\) the author raises the question: Does there exist a spline \(s\in Sp(k+1,j,\Delta_ n)\) satisfying the conditions: \(s(x_ i)=y_ i\) \((i=0,1,...,n\); \(y_ i\) given reals), such that s is convex and increasing on the interval [a,b]? In this paper the author considers only the case when \(1\leq j\leq [(k+1)/2]\). Sufficient conditions which guarantee the existence of s and an explicit formula for this function are derived in detail. It should be mentioned that closely related to the author’s results are those given earlier by D. F. McAllister and J. A. Roulier [Math. Comput. 32, 1154-1162 (1978; Zbl 0398.41004)].
Reviewer: D.D.Stancu

MSC:

41A15 Spline approximation
41A05 Interpolation in approximation theory
65D07 Numerical computation using splines
65D10 Numerical smoothing, curve fitting
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