On interpolation with connected polynomials


We consider a function \(f\in C^{n-1}[a,b]\) and the nodes \(a=x_{0}<x_{1}<\ldots<x_{m}=b\). Given the values \begin{align}f\left( x_{i}\right) =&u_{i}, \quad i=1,…,m  \\ f^{\left( k\right) }\left( x_0\right) =&p_{0}^{k},\quad k=1,…,n-1,\end{align} we show that there exists on each interval \(\left[ x_{i-1},x_{i}\right] ,\ i=1,…,m,\) a unique polynomial \(P_{i}\) of degree \(m\) that satisfies \begin{align} P\left( x_{i}\right) =& u_{i},  \\ P_{i}^{\left( k\right) }\left( x_{i}\right) =&P_{i-1}^{\left( k\right) }\left(x_{i}\right) ,\quad k=1,…,n-1,\end{align} i.e. the resulted polynomials are successively joined. In this paper we show how these polynomials may be constructed. The functions \(f\) is therefore approximated by these polynomials.

Original Title (in French)


Ion Păvăloiu – Tiberiu Popoviciu Institute of Numerical Analysis


approximation by polynomials, joined polynomials, Hermite interpolation


[1] Boor Carl de, Bicubic sþIine interpolation. Journal of Mathenatics and Physics. 41 (1962) 3.

[2]                 I959.


About this paper

Cite this paper as:

I. Păvăloiu, Sur l’intérpolations à l’aide des polynômes raccordées, Mathematica, 6(27) (1964) no. 2, pp. 295-299 (in French).



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Editura Academiei R.S. Române


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