On interpolation with connected polynomials

Abstract

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.

Authors

Ion Păvăloiu – Tiberiu Popoviciu Institute of Numerical Analysis

Keywords

approximation by polynomials, joined polynomials, Hermite interpolation

References

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

[2]                 I959.

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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).

Mathematica

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