Difference between revisions of "Bilateral Laplace transform"
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Let $\mathbb{T}$ be a [[time scale]]. The Bilateral Laplace transform of a function $f \colon \mathbb{T} \rightarrow \mathbb{T}$ centered at $s$ is given by | Let $\mathbb{T}$ be a [[time scale]]. The Bilateral Laplace transform of a function $f \colon \mathbb{T} \rightarrow \mathbb{T}$ centered at $s$ is given by | ||
| − | $$ | + | $$\mathscr{L}_{\mathbb{T}}^b(z;s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t,$$ |
| − | + | where $e_{\ominus z}$ denotes the [[delta exponential]] and $\ominus z$ denotes [[forward circle minus]]. | |
| + | |||
| + | =See also= | ||
| + | [[Unilateral Laplace transform]]<br /> | ||
| + | [[Cuchta-Georgiev Fourier transform]]<br /> | ||
=References= | =References= | ||
| − | *{{PaperReference|Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series|2009|John M. Davis|author2=Ian A. Gravagne|author3=Robert J. Marks II|prev=findme|next=findme}}: | + | *{{PaperReference|Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series|2009|John M. Davis|author2=Ian A. Gravagne|author3=Robert J. Marks II|prev=findme|next=findme}}: $(3.1)$ |
| + | *{{PaperReference|Analysis of the bilateral Laplace transform on time scales with applications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Section 1 | ||
| + | |||
| + | [[Category:Definition]] | ||
Latest revision as of 15:12, 21 January 2023
Let $\mathbb{T}$ be a time scale. The Bilateral Laplace transform of a function $f \colon \mathbb{T} \rightarrow \mathbb{T}$ centered at $s$ is given by $$\mathscr{L}_{\mathbb{T}}^b(z;s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t,$$ where $e_{\ominus z}$ denotes the delta exponential and $\ominus z$ denotes forward circle minus.
See also
Unilateral Laplace transform
Cuchta-Georgiev Fourier transform
References
- John M. Davis, Ian A. Gravagne and Robert J. Marks II: Bilateral Laplace Transforms on Time Scales: Convergence, Convolution, and the Characterization of Stationary Stochastic Time Series (2009)... (previous)... (next): $(3.1)$
- Tom Cuchta and Svetlin Georgiev: Analysis of the bilateral Laplace transform on time scales with applications (2021): Section 1
