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
$$F(z,s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t.$$
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$$\mathscr{L}_{\mathbb{T}}^b(z;s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t,$$
This integral is clearly a generalization of the [[Laplace transform]].
+
where $e_{\ominus z}$ denotes the [[delta exponential]] and $\ominus z$ denotes [[forward circle minus]].
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=See also=
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[[Unilateral Laplace transform]]<br />
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[[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}}: Definition $1.1$, $(1.1)$
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*{{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}}: $(3.1)$
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*{{PaperReference|Analysis of the bilateral Laplace transform on time scales with applications|2021|Tom Cuchta|author2=Svetlin Georgiev|prev=|next=}}: Section 1
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[[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