Bilateral Laplace transform

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 $$F(z,s)=\displaystyle\int_{-\infty}^{\infty} f(t)e_{\ominus z}(\sigma(t),s) \Delta t.$$ This integral is clearly a generalization of the Laplace 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