Piecewise convex interpolation

Let \(n\in N\) and the following two systems of \(n+1\) real values:
\[
0=x_{0}<x_{1}<\ldots<x_{n}=1
\]
\[
0=y_{0},\ y_{1},\ \ldots,\ y_{n}.
\]
In the papers [1], [2] it is proved that if \(n\geq1\) and \(y_{i}-y_{i-1}\neq0,\ i=1,2,\ldots,n\) then there exists a polynomial \(P\) which assumes at each point \(x_{i}\) the preassigned value \(y_{i}\) and which is piecewise monotone, more precisely:
\[
P\left( x_{i}\right) =y_{i},\ \ i=0,1,\ldots,n
\]
\[
P^{\prime}\left( x\right) \left( y_{i}-y_{i-1}\right) \geq0,\ \ \ x\in
\left[ x_{i-1},x_{i}\right] ,\ \ i=1,2,\ldots,n.
\]
There are many papers related to the piecewise monotone interpolation; such references can be found in [4], [5].

The purpose of this paper is to prove the existence of a piecewise convex (by order \(p=1\)) interpolating polynomial. Our proof uses the Wolibner-Young’s theorem [1], [2] concerning the piecewise monotone (convex by order \(p=0\)) interpolation, in the same way that the last one uses the Weierstrass approximation theorem.

Radu Precup
Liceul de Informatică, Cluj-Napoca

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https://ictp.acad.ro/jnaat/journal/article/view/1985-vol14-no2-art4/1114

R. Precup, Piecewise convex interpolation, Rev. Anal. Numér. Théor. Approx., 14 (1985) no. 2, pp. 123-126.

MR: 87m:41004.

citations

[1] Wolibner, W., Sur un polynôme d’interpolation. (French) Colloquium Math. 2, (1951). 136-137, MR0043946 .

[2] Young, S., Piecewise monotone interpolation, Bull. Amer. Math. Soc., 73, 642-643.

[3] Nikolčeva, M.G., Interpolation of monotone and convex functions, Proceedings of the International Conference on constructive function Theory, Blagcevgrad, May 30 – June 6, 1977, Sofia 1980, 437-442.

[4] Iliev, G. L., Exact estimates for monotone interpolation. J. Approx. Theory 28 (1980), no. 2, 101-112, MR0573325.

[5] Precup, Radu, Estimates of the degree of comonotone interpolating polynomials. Anal. Numér. Théor. Approx. 11 (1982), no. 1-2, 139-145, MR0692479.