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.

Original Title (in French)

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).

Journal

Mathematica

Publisher Name

Editura Academiei R.S. Române

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