## Abstract

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.

## Authors

Radu **Precup****
**Liceul de Informatică, Cluj-Napoca

## Keywords

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##### Cite this paper as:

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

## About this paper

##### Journal

Mathematica – Revue d’analyse numérique et de théorie d’approximation

##### Publisher Name

Editions de l’Academie Roumaine

##### paper on journal website

##### Print ISSN

Not available yet.

##### Online ISSN

Not available yet.

MR: 87m:41004.

##### Google Scholar citations

## References

[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.