Dual of the function algebra \(A^{-\infty }(D)\) and representation of functions in Dirichlet series. (English) Zbl 1205.32004
Let \(D\) be a bounded \(C^2\)-smooth convex domain in \(\mathbb C^n\) and let \(d(z)\) be the distance function of \(z\) to the boundary of \(D\). The authors determine the dual, \(A^{-\infty}_D\), of the space
\[
A^{-\infty}(D)=\{f \text{ holomorphic in } D: \exists n\in \mathbb N, \sup_{z\in D} |f(z)|\, d(z)^n<\infty\},
\]
endowed with a natural inductive limit topology. Countable sets of sufficiency for \(A^{-\infty}_D\) are constructed. Discussed is also the problem of representing functions in \(A^{-\infty}(D)\) by Dirichlet series \(\sum_{k=1}^\infty c_k e^{\left <\lambda_k,z\right>}\).
Reviewer: Raymond Mortini (Metz)
MSC:
| 32A38 | Algebras of holomorphic functions of several complex variables |
| 46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |
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