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)

Sur l’intérpolation à l’aide des polynômes raccordées

Authors

Ion Păvăloiu
Tiberiu Popoviciu Institute of Numerical Analysis

Keywords

approximation by polynomials, joined polynomials, Hermite interpolation

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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).

About this paper

Journal

Mathematica

Publisher Name

Editura Academiei R.S. Române

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References

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

[2]                 I959.

1962

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