IJPAM: Volume 86, No. 1 (2013)

A NOTE ON TWO POINT TAYLOR EXPANSION II

Kazuaki Kitahara1, Takako Yamada2, Kazuki Fujiwara3
1,3School of Science and Technology
Kwansei Gakuin University
Sanda, 669-1337, JAPAN
2School of Policy Studies
Kwansei Gakuin University
Sanda, 669-1337, JAPAN


Abstract. Let $f$ be a continuous function on $[-r, r]$ $(r > 1+\sqrt{2})$, $\alpha $ the function $f$ restricted to the subinterval $[0, r]$ and $\beta $ the function $f$ restricted to the subinterval $[-r, 0]$. If $\alpha ~({\rm resp.} \beta )$ is expressed as the Taylor expansion of $\alpha ~({\rm resp.} \beta )$ about $1~ ({\rm resp.} -1)$, then we show that $f$ is expressed as the two point Taylor expansion about $-1, 1$ on the interval $(-\sqrt{2}, \sqrt{2})$. Furthermore, the $k$-th order derivatives of $f$ on $(-\sqrt{2}, 0) \cup (0, \sqrt{2})$ are expressed as the termwise $k$ times differentiation of the two point Taylor expansion about $-1, 1$.

Received: March 3, 2013

AMS Subject Classification: 41A10, 41A05, 41A58

Key Words and Phrases: Hermite interpolation, Taylor expansion, termwise differentiation

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DOI: 10.12732/ijpam.v86i1.7 How to cite this paper?
Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 86
Issue: 1