## 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

### Citations:

Zbl 0436.41001; Zbl 0398.41004
Full Text:

### References:

 [1] H. B. Curry and I. J. Schoenberg,On Pólya frequency functions IV. The fundamental spline functions and their limits, J. Anal. Math. 17 (1966), 71–107. · Zbl 0146.08404 [2] T. N. E. Greville,Introduction to spline functions, in:Theory and Applications of Spline Functions (T.N.E. Greville, Ed.), 1–36, New York: Academic Press, 1969. · Zbl 0215.17601 [3] D. F. McAllister and J. A. Roulier,Interpolation by convex quadratic splines, Math. Comp. 32 (1978), 1154–1162. · Zbl 0398.41004 [4] D. C. Myers and J. A. Roulier,Markov-type inequalities and the degree of convex spline interpolation, J. Approximation Theory 28 (1980), 267–272. · Zbl 0459.41005 [5] E. Neuman,Convex interpolation splines of odd degree, Utilitas Math. 14 (1978), 129–140. · Zbl 0393.41005 [6] E. Neuman,Uniform approximation by some Hermite interpolating splines, J. Comput. Appl. Math. 4 (1978), 7–9. · Zbl 0388.41007 [7] E. Neuman,Convex interpolating splines of arbitrary degree, in:Numerical Methods of Approximation Theory (L. Collatz, G. Meinardus and H. Werner, Eds.), 211–222, Basel: Birkhäuser Verlag, 1980. [8] E. Neuman,Properties of a class of polynomial splines (submitted for publication). [9] E. Passow and J. A. Roulier,Shape preserving spline interpolation, in:Approximation Theory 11 (G. G. Lorentz, C. Chui and L. L. Schumaker, Eds.), 503–507, New York: Academic Press, 1976. · Zbl 0354.41002 [10] E. Passow and J. A. Roulier,Monotone and convex spline interpolation, SIAM J. Numer. Anal. 14 (1977), 904–909. · Zbl 0378.41002 [11] L. L. Schumaker,Spline Functions: Basic Theory, New York: John Wiley and Sons, 1981. · Zbl 0449.41004
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