月曜解析セミナー： Representation of functions in \(A^{-\infty}\) by exponential series and applications

開催日時

2018年10月1日 15時00分 ～ 2018年10月1日 16時30分

場所

理学部3号館210

講演者

Le Hai Khoi (Nanyang Technological University, Singapole)

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.