# The Laplace derivative

Commentationes Mathematicae Universitatis Carolinae (2001)

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

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topSvetic, 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 -

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