R.E. Svetic
The Laplace derivative

Comment.Math.Univ.Carolinae 42,2 (2001) 331-343.

Abstract:A function $f:\Bbb R \rightarrow \Bbb R$ 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 $f_{\delimiter "426830A n\delimiter "526930B }(x)$. \par 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.

Keywords: Peano derivative, generalized Peano derivative, Laplace derivative, Laplace transform, Tauberian theorem
AMS Subject Classification: Primary 26A24; Secondary 26A21, 26A48, 40E05, 44A10