How is the inverse Laplace-Borel transform defined?
Si $\mu\in (\mathcal{O}(\mathbb{C}))^{\ast}$ (The dual space of the entire functions or analytic functionals) then the Laplace-Borel $\mathcal{L}\mu(\xi)\equiv \langle \mu(z),\mathrm{e}^{z\xi}\rangle$ is an isomorphism between $(\mathcal{O}(\mathbb{C}))^{\ast}$ and $\text{Exp}(\mathbb{C})$ (the entire functions of exponential type). If $f(z)\in \text{Exp}(\mathbb{C})$. How is $\mathcal{L}^{-1}f(z)$ defined? What analytical functional is it?
Thanks.
Refer to Treves "Topological vector spaces, distributions and kernels". See also Borel-Fourier transform.
Analytic functional is a continuous linear functional on the space of entire functions. Textbook provides details re. topologies on relevant spaces etc.
There is reverse formula provided: $$ f(\xi) = \langle\mu_f(z), e^{z\xi}\rangle\in \mathrm{Exp(\mathbb{C}}) $$ where $\mu_f (F) = \sum_n a_n f^{(n)}(0)$ given $F(z) = \sum_n a_n z^n\in \mathcal{O}(\mathbb{C}).$