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 | ||
$$\mathscr{L}_{\mathbb{T}}^b(z;s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t,$$ | $$\mathscr{L}_{\mathbb{T}}^b(z;s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t,$$ | ||
| − | where $e_{\ominus z}$ denotes the [[delta exponential]]. | + | where $e_{\ominus z}$ denotes the [[delta exponential]] and $\ominus z$ denotes [[forward circle minus]]. |
=See also= | =See also= | ||
Revision as of 12:50, 16 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
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
