Monday Analysis Seminar: Representation of functions in $A^{-\infty}$ by exponential series and applications

Date
2018-10-1 15:00 - 2018-10-1 16:30
Place
Faculty of Science Building #3 Room 210
Speaker/Organizer
Le Hai Khoi (Nanyang Technological University, Singapore)
 
Let $\Omega$ be a bounded convex domain in $\mathbb{C}^n\ (n\ge 1)$ and $\displaystyle d(z) := \inf_{\zeta\in\partial\Omega}|z-\zeta|$, $z\in\Omega$. The space $A^{-\infty}(\Omega)$ of holomorphic functions in $\Omega$ with polynomial growth near the boundary $\partial\Omega$, equipped with its natural inductive limit topology, is defined as
\[
A^{-\infty}(\Omega) := \left\{f\in{\mathcal{O}}(\Omega) : \exists \ p > 0, ~~ \sup_{z\in\Omega} |f(z)|~[d(z)]^p <\infty \right\}.
\]
This function algebra, as is well-known, arises from Schwartz theory of distributions.

Our talk, which is based on joint works with A.Abanin and R. Ishimura, is concerned with a question: Is it possible to represent functions from $A^{-\infty}(\Omega)$ in a form of Dirichlet (exponential) series
\[
f(z) = \sum_{k=1}^\infty c_ke^{\langle\lambda_k,z\rangle}, \quad z\in\Omega,
\]
that converges for the topology of this space? The applications to functional equations are discussed. Open problems for further study are also given.

Place: Science Building #3 Room 210

Monday Analysis Seminar