# The Laplace derivative

• Volume: 42, Issue: 2, page 331-343
• ISSN: 0010-2628

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## Abstract

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A function $f:ℝ\to ℝ$ is said to have the $n$-th Laplace derivative on the right at $x$ if $f$ is continuous in a right neighborhood of $x$ and there exist real numbers ${\alpha }_{0},...,{\alpha }_{n-1}$ such that ${s}^{n+1}{\int }_{0}^{\delta }{e}^{-st}\left[f\left(x+t\right)-{\sum }_{i=0}^{n-1}{\alpha }_{i}{t}^{i}/i!\right]\phantom{\rule{0.166667em}{0ex}}dt$ converges as $s\to +\infty$ for some $\delta >0$. There is a corresponding definition on the left. The function is said to have the $n$-th Laplace derivative at $x$ when these two are equal, the common value is denoted by ${f}_{〈n〉}\left(x\right)$. In this work we establish the basic properties of this new derivative and show that, by an example, it is more general than the generalized Peano derivative; hence the Laplace derivative generalizes the Peano and ordinary derivatives.

## How to cite

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Svetic, Ralph E.. "The Laplace derivative." Commentationes Mathematicae Universitatis Carolinae 42.2 (2001): 331-343. <http://eudml.org/doc/248768>.

@article{Svetic2001,
abstract = {A function $f:\mathbb \{R\} \rightarrow \mathbb \{R\}$ is said to have the $n$-th Laplace derivative on the right at $x$ if $f$ is continuous in a right neighborhood of $x$ and there exist real numbers $\alpha _0, \ldots , \alpha _\{n-1\}$ such that $s^\{n+1\}\int _0^\delta e^\{-st\}[f(x+t)-\sum _\{i=0\}^\{n-1\}\alpha _i t^i/i!]\,dt$ converges as $s\rightarrow +\infty$ for some $\delta >0$. There is a corresponding definition on the left. The function is said to have the $n$-th Laplace derivative at $x$ when these two are equal, the common value is denoted by $f_\{\langle n\rangle \}(x)$. In this work we establish the basic properties of this new derivative and show that, by an example, it is more general than the generalized Peano derivative; hence the Laplace derivative generalizes the Peano and ordinary derivatives.},
author = {Svetic, Ralph E.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Peano derivative; generalized Peano derivative; Laplace derivative; Laplace transform; Tauberian theorem; Laplace transform; Laplace derivative; generalized Peano derivative},
language = {eng},
number = {2},
pages = {331-343},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The Laplace derivative},
url = {http://eudml.org/doc/248768},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Svetic, Ralph E.
TI - The Laplace derivative
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 2
SP - 331
EP - 343
AB - A function $f:\mathbb {R} \rightarrow \mathbb {R}$ is said to have the $n$-th Laplace derivative on the right at $x$ if $f$ is continuous in a right neighborhood of $x$ and there exist real numbers $\alpha _0, \ldots , \alpha _{n-1}$ such that $s^{n+1}\int _0^\delta e^{-st}[f(x+t)-\sum _{i=0}^{n-1}\alpha _i t^i/i!]\,dt$ converges as $s\rightarrow +\infty$ for some $\delta >0$. There is a corresponding definition on the left. The function is said to have the $n$-th Laplace derivative at $x$ when these two are equal, the common value is denoted by $f_{\langle n\rangle }(x)$. In this work we establish the basic properties of this new derivative and show that, by an example, it is more general than the generalized Peano derivative; hence the Laplace derivative generalizes the Peano and ordinary derivatives.
LA - eng
KW - Peano derivative; generalized Peano derivative; Laplace derivative; Laplace transform; Tauberian theorem; Laplace transform; Laplace derivative; generalized Peano derivative
UR - http://eudml.org/doc/248768
ER -

## References

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12. Mukhopadhyay S.N., Mitra S., Measurability of Peano derivatives and approximate Peano derivatives, Real Anal. Exchange 20 (1994-5), 768-775. (1994-5) MR1348098
13. Oliver H.W., The exact Peano derivative, Trans. Amer. Math. Soc. 76 (1954), 444-456. (1954) Zbl0055.28505MR0062207
14. Svetic R.E., Volkmer H., On the ultimate Peano derivative, J. Math. Anal. Appl. 218 (1998), 439-452. (1998) Zbl0893.26001MR1605380
15. Verblunsky S., On the Peano derivatives, Proc. London Math. Soc. 22 (1971), 313-324. (1971) Zbl0209.36401MR0285678
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17. Widder D.V., An Introduction to Transform Theory, Academic Press, New York, 1971. Zbl0219.44001

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